Existence of solutions for some non-Fredholm integro
Transcript Of Existence of solutions for some non-Fredholm integro
Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion
Messoud Efendiev1,2, Vitali Vougalter3 1 Helmholtz Zentrum Mu¨nchen, Institut fu¨r Computational Biology, Ingolsta¨dter Landstrasse 1
Neuherberg, 85764, Germany e-mail: [email protected] 2 Department of Mathematics, Marmara University, Istanbul, Turkey
e-mail: [email protected]
3 Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada e-mail: [email protected]
Abstract. We establish the existence in the sense of sequences of solutions for certain integro-differential type equations in two dimensions involving the normal diffusion in one direction and the anomalous diffusion in the other direction in H2(R2) via the fixed point technique. The elliptic equation contains a second order differential operator without the Fredholm property. It is proved that, under the reasonable technical conditions, the convergence in L1(R2) of the integral kernels implies the existence and convergence in H2(R2) of the solutions.
Keywords: solvability conditions, non Fredholm operators, integro-differential equations, mixed diffusion AMS subject classification: 35J05, 35P30, 47F05
1 Introduction
Let us recall that a linear operator L acting from a Banach space E into another Banach space F satisfies the Fredholm property if its image is closed, the dimension of its kernel and the codimension of its image are finite. As a consequence, the equation Lu = f is solvable if and only if φi(f ) = 0 for a finite number of functionals φi from the dual space F ∗. These properties of Fredholm operators are widely used in many methods of linear and nonlinear analysis.
Elliptic problems in bounded domains with a sufficiently smooth boundary satisfy the Fredholm property if the ellipticity condition, proper ellipticity and Shapiro-Lopatinskii conditions are fulfilled (see e.g. [1], [6], [16], [20]). This is the main result of the theory of linear
1
elliptic equations. In the case of unbounded domains, these conditions may not be sufficient and the Fredholm property may not be satisfied. For instance, the Laplace operator, Lu = ∆u, in Rd does not satisfy the Fredholm property when considered in H¨older spaces, L : C2+α(Rd) → Cα(Rd), or in Sobolev spaces, L : H2(Rd) → L2(Rd).
Linear elliptic problems in unbounded domains satisfy the Fredholm property if and only if, in addition to the conditions cited above, limiting operators are invertible (see [21]). In some simple cases, limiting operators can be explicitly constructed. For instance, if
Lu = a(x)u′′ + b(x)u′ + c(x)u, x ∈ R,
where the coefficients of the operator have limits at infinity,
a± = lim a(x), x→±∞
the limiting operators are:
b± = lim b(x), x→±∞
c± = lim c(x), x→±∞
L±u = a±u′′ + b±u′ + c±u.
Since the coefficients are constants, the essential spectrum of the operator, that is the set of complex numbers λ for which the operator L − λ fails to satisfy the Fredholm property, can be explicitly found by virtue of the Fourier transform:
λ±(ξ) = −a±ξ2 + b±iξ + c±, ξ ∈ R.
Invertibility of limiting operators is equivalent to the condition that the essential spectrum
does not contain the origin.
In the case of general elliptic problems, the same assertions hold true. The Fredholm
property is satisfied if the essential spectrum does not contain the origin or if the limiting
operators are invertible. However, these conditions may not be explicitly written.
In the case of non-Fredholm operators the usual solvability conditions may not be ap-
plicable and solvability conditions are, in general, not known. There are some classes of
operators for which solvability conditions are derived. We illustrate them with the following
example. Consider the problem
Lu ≡ ∆u + au = f
(1.1)
in Rd, where a is a positive constant. The operator L coincides with its limiting operators.
The homogeneous equation has a nonzero bounded solution. Hence the Fredholm property is
not satisfied. However, since the operator has constant coefficients, we can apply the Fourier
transform and find the solution explicitly. Solvability relations can be formulated as follows.
If f ∈ L2(Rd) and xf ∈ L1(Rd), then there exists a solution of this problem in H2(Rd) if
and only if
f (x), eipx
= 0, p ∈ S√d a.e.
d
(2π) 2 L2(Rd)
a
(see [26]). Here and further down Srd stands for the sphere in Rd of radius r centered at the origin. Thus, though the operator does not satisfy the Fredholm property, solvability
2
relations are formulated similarly. However, this similarity is only formal since the range of the operator is not closed.
In the case of the operator with a potential,
Lu ≡ ∆u + a(x)u = f,
Fourier transform is not directly applicable. Nevertheless, solvability relations in R3 can be derived by a rather sophisticated application of the theory of self-adjoint operators (see [24]). As before, solvability conditions are formulated in terms of orthogonality to solutions of the homogeneous adjoint equation. There are several other examples of linear elliptic non Fredholm operators for which solvability conditions can be obtained (see [11], [21], [22], [23], [24], [26]).
Solvability relations play a crucial role in the analysis of nonlinear elliptic problems. In the case of non-Fredholm operators, in spite of some progress in understanding of linear equations, there exist only few examples where nonlinear non-Fredholm operators are analyzed (see [5], [9], [25], [26], [29]). The large time behavior of solutions of a class of fourth-order parabolic equations defined on unbounded domains using the Kolmogorov ε-entropy as a measure was studied in [8]. The work [7] deals with the finite and infinite dimensional attractors for evolution equations of mathematical physics. The attractor for a nonlinear reaction-diffusion system in an unbounded domain in R3 was investigated in [12]. The articles [13] and [19] are devoted to the understanding of the Fredholm and properness properties of quasilinear elliptic systems of second order and of operators of this kind on RN . Exponential decay and Fredholm properties in second-order quasilinear elliptic systems were addressed in [14]. In the present article we treat another class of stationary nonlinear problems, for which the Fredholm property may not be satisfied:
∂2u ∂x2 −
1
∂2 s
− ∂x2 u + G(x − y)F (u(y), y)dy = 0,
2
R2
0 < s < 1,
(1.2)
where x = (x1, x2) ∈ R2, y = (y1, y2) ∈ R2. Here the operator
∂2 Ls := − ∂x2 +
1
∂2 − ∂x2
2
s
: H2(R2) → L2(R2),
0
(1.3)
is defined via the spectral calculus. The novelty of the present work is that in the diffusion term we add the standard minus Laplacian in the x1 variable with the negative Laplace operator in x2 raised to a fractional power. Such model is new and not much is understood about it, especially in the context of the integro-differential equations. The difficulty we have to overcome is that such problem becomes anisotropic and it is more technical to obtain the desired estimates when dealing with it. In population dynamics in the Mathematical Biology the integro-differential problems describe models with intra-specific competition and nonlocal consumption of resources (see e.g. [2], [3]). It is important to study the equations of this kind in unbounded domains from the point of view of the understanding of the spread of the viral infections, since many countries have to deal with the pandemics. We use the explicit
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form of the solvability conditions and establish the existence of solutions of such nonlinear equation. In the case of the standard Laplacian instead of (1.3), the problem analogous to (1.2) was considered in [25] and [29]. The solvability of the integro-differential equations involving in the diffusion term only the negative Laplace operator raised to a fractional power was actively studied in recent years in the context of the anomalous diffusion (see e.g. [10], [27], [28]). The anomalous diffusion can be described as a random process of particle motion characterized by the probability density distribution of jump length. The moments of this density distribution are finite in the case of the normal diffusion, but this is not the case for the anomalous diffusion. The asymptotic behavior at infinity of the probability density function determines the value of the power of the Laplacian (see [18]). In article [17] the authors prove the imbedding theorems and study the spectrum of certain pseudodifferential operators.
2 Formulation of the results
The nonlinear part of equation (1.2) will satisfy the following regularity conditions.
Assumption 1. Function F (u, x) : R × R2 → R is satisfying the Caratheodory condition (see [15]), such that
|F (u, x)| ≤ k|u| + h(x) f or u ∈ R, x ∈ R2
(2.1)
with a constant k > 0 and h(x) : R2 → R+, h(x) ∈ L2(R2). Moreover, it is a Lipschitz continuous function, such that
|F (u1, x) − F (u2, x)| ≤ l|u1 − u2| f or any u1,2 ∈ R, x ∈ R2
(2.2)
with a constant l > 0.
The solvability of a local elliptic equation in a bounded domain in RN was considered in [4], where the nonlinear function was allowed to have a sublinear growth. In order to study of the existence of solutions of (1.2), we introduce the auxiliary equation
∂2u − ∂x2 +
1
∂2 s
− ∂x2 u = G(x − y)F (v(y), y)dy,
2
R2
0 < s < 1.
(2.3)
We denote
(f1(x), f2(x))L2(R2) := f1(x)f¯2(x)dx,
R2
(2.4)
with a slight abuse of notations when these functions are not square integrable. Indeed,
if f1(x) ∈ L1(R2) and f2(x) is bounded, like for instance those involved in orthogonality
relation (4.4) below, the integral in the right side of (2.4) makes sense. In the the article we
work in the space of the two dimensions, such that the appropriate Sobolev space is equipped
with the norm
u
2 H 2 (R2 )
:=
u
2 L2 (R2 )
+
∆u
. 2
L2 (R2 )
(2.5)
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In the equation above we are dealing with the operator Ls defined in (1.3). By virtue of
the standard Fourier transform (4.1), it can be easily checked that its essential spectrum is
given by
λs(p) = p21 + |p2|2s, p = (p1, p2) ∈ R2.
(2.6)
Since set (2.6) contains the origin, our operator Ls fails to satisfy the Fredholm property, which is the obstacle to solve our equation.
The similar situations but in linear problems, both self- adjoint and non self-adjoint involving non Fredholm differential operators have been studied extensively in recent years (see [21], [22], [24], [26]). Our present work is related to our article [11] since we also deal with the non Fredholm operator, now involved in the problem, which is not linear anymore and contains the nonlocal terms. Currently, as distinct from [11], the space dimension is restricted to d = 2 to avoid the extra technicalities.
In the present work we manage to establish that under the reasonable technical assumptions problem (2.3) defines a map T2, s : H2(R2) → H2(R2), 0 < s < 1, which is a strict contraction.
Theorem 1. Let Assumption 1 hold, 0 < s < 1, the function G(x) : R2 → R, such that G(x) ∈ L1(R2) and x2G(x) ∈ L1(R2). Moreover, (−∆)1−sG(x) ∈ L1(R2).
We also assume that orthogonality conditions (4.4), (4.5) hold if 0 < s ≤ 1 and relations
1
√
2
(4.4), (4.5) and (4.6) are valid for 2 < s < 1 and that 2 2πN2, sl < 1. Then the map
T2,sv = u on H2(R2) defined by problem (2.3) admits a unique fixed point v2,s, which is the
only solution of equation (1.2) in H2(R2).
This fixed point v2,s is nontrivial provided the intersection of supports of the Fourier transforms of functions suppF (0, x) ∩ suppG is a set of nonzero Lebesgue measure in R2.
Related to equation (1.2) in the space of two dimensions, we study the sequence of approximate equations with m ∈ N
∂2um ∂x2 −
1
∂2 s
− ∂x2 um + Gm(x − y)F (um(y), y)dy = 0,
2
R2
0 < s < 1.
(2.7)
The sequence of kernels {Gm(x)}∞ m=1 tends to G(x) as m → ∞ in the appropriate function spaces discussed below. We will show that, under the appropriate technical conditions, each of equations (2.7) has a unique solution um(x) ∈ H2(R2), the limiting problem (1.2) admits a unique solution u(x) ∈ H2(R2), and um(x) → u(x) in H2(R2) as m → ∞, which is the so-called existence of solutions in the sense of sequences. In this case, the solvability relations
can be formulated for the iterated kernels Gm. They yield the convergence of the kernels in terms of the Fourier transforms (see the Appendix) and, as a consequence, the convergence
or the solutions (Theorem 2 below). Similar ideas in the context of the standard Schro¨dinger
type operators were exploited in [23]. Our second main proposition is as follows.
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Theorem 2. Let Assumption 1 hold, 0 < s < 1, m ∈ N, the functions Gm(x) : R2 → R are such that Gm(x) ∈ L1(R2), x2Gm(x) ∈ L1(R2) and (−∆)1−sGm(x) ∈ L1(R2). Moreover, Gm(x) → G(x) in L1(R2), x2Gm(x) → x2G(x) and (−∆)1−sGm(x) → (−∆)1−sG(x) in L1(R2) as m → ∞. We also assume that for all m ∈ N orthogonality conditions (4.21), (4.22) hold if 0 < s ≤ 1
12 and relations (4.21), (4.22) and (4.23) are valid for 2 < s < 1. Furthermore, we suppose that (4.24) holds for all m ∈ N with a certain 0 < ε < 1.
Then each problem (2.7) admits a unique solution um(x) ∈ H2(R2), limiting equation (1.2) possesses a unique solution u(x) ∈ H2(R2) and um(x) → u(x) in H2(R2) as m → ∞.
The unique solution um(x) of each problem (2.7) is nontrivial provided that the intersection of supports of the Fourier transforms of functions suppF (0, x) ∩ suppGm is a set of nonzero Lebesgue measure in R2. Similarly, the unique solution u(x) of limiting equation (1.2) does not vanish identically if suppF (0, x) ∩ suppG is a set of nonzero Lebesgue measure in R2.
Remark 1. In the article we work with real valued functions by virtue of the assumptions on F (u, x), Gm(x) and G(x) involved in the nonlocal terms of the iterated and limiting problems discussed above.
Remark 2. The importance of Theorem 2 above is the continuous dependence of solutions with respect to the integral kernels.
3 Proofs Of The Main Results
Proof of Theorem 1. Let us first suppose that for a certain v(x) ∈ H2(R2) there exist two
solutions u1,2(x) ∈ H2(R2) of problem (2.3). Then their difference w(x) := u1(x) − u2(x) ∈ H2(R2) will be a solution of the homogeneous equation
∂2w − ∂x2 +
1
∂2 − ∂x2
2
s
w = 0.
Because the operator Ls : H2(R2) → L2(R2) defined in (1.3) does not have any nontrivial zero modes, the function w(x) vanishes in the space of two dimensions.
We choose arbitrarily v(x) ∈ H2(R2). Let us apply the standard Fourier transform (4.1)
to both sides of (2.3) and arrive at
G(p)f (p) u(p) = 2π p2 + |p2|2s ,
1
p2u(p) = 2π p2G(p)f (p), p21 + |p2|2s
(3.1)
where f (p) denotes the Fourier image of F (v(x), x). Evidently, we have the estimates from
above |u(p)| ≤ 2πN2, s|f (p)| and |p2u(p)| ≤ 2πN2, s|f (p)|.
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Note that N2, s < ∞ by means of Lemma 3 of the Appendix under the given conditions. This enables us to obtain the upper bound on the norm
u
2 H 2 (R2 )
=
u(p)
2 L2 (R2 )
+
p2u(p)
2 L2 (R2 )
≤
8π2N22,
s
F (v(x), x)
, 2
L2 (R2 )
which is finite by virtue of (2.1) of Assumption 1 because v(x) ∈ L2(R2). Clearly, v(x) ∈ H2(R2) ⊂ L∞(R2) due to the Sobolev embedding. Thus, for an arbitrary v(x) ∈ H2(R2) there exists a unique solution u(x) ∈ H2(R2) of problem (2.3), such that its Fourier image is given by (3.1). Hence, the map T2,s : H2(R2) → H2(R2) is well defined. This allows us to choose arbitrary functions v1,2(x) ∈ H2(R2), such that their images u1,2 := T2,sv1,2 ∈ H2(R2).
Clearly, (2.3) yields
∂2u1 − ∂x2 +
1
∂2 s
− ∂x2 u1 = G(x − y)F (v1(y), y)dy,
2
R2
(3.2)
− ∂2u2 + ∂x21
∂2 s
− ∂x2 u2 = G(x − y)F (v2(y), y)dy,
2
R2
(3.3)
where 0 < s < 1. Let us apply the standard Fourier transform (4.1) to both sides of the equations of system (3.2), (3.3) above. We arrive at
u1(p) = 2π G(p)f1(p) , p21 + |p2|2s
u2(p) = 2π G(p)f2(p) . p21 + |p2|2s
(3.4)
Here f1(p) and f2(p) stand for the Fourier images of F (v1(x), x) and F (v2(x), x) respectively. By means of (3.4) we derive the upper bounds
|u1(p) − u2(p)| ≤ 2πN2, s|f1(p) − f2(p)|, p2|u1(p) − u2(p)| ≤ 2πN2, s|f1(p) − f2(p)|,
such that
u1 − u2
2 H 2 (R2 )
=
u1(p) − u2(p)
2 L2 (R2 )
+
p2[u1(p) − u2(p)]
2 L2 (R2 )
≤
≤ 8π2N22, s
F (v1(x), x) − F (v2(x), x)
. 2
L2 (R2 )
Evidently, v1,2(x) ∈ H2(R2) ⊂ L∞(R2) via the Sobolev embedding. Condition (2.2) above
implies that
√ T2,sv1 − T2,sv2 H2(R2) ≤ 2 2πN2, sl v1 − v2 H2(R2)
and the constant in the right side of this inequality is less than one via the one of our
assumptions. Thus, by means of the Fixed Point Theorem, there exists a unique function v2,s ∈ H2(R2) with the property T2,sv2,s = v2,s, which is the only solution of problem (1.2) in H2(R2). Suppose v2,s(x) = 0 identically in the space of two dimensions. This will contradict to our assumption that the Fourier images of G(x) and F (0, x) do not vanish on a set of nonzero Lebesgue measure in R2.
7
Let us proceed to establishing the solvability in the sense of sequences for our integrodifferential problem in the space of two dimensions.
Proof of Theorem 2. By virtue of the result of Theorem 1 above, each problem (2.7) has a unique solution um(x) ∈ H2(R2), m ∈ N. Limiting equation (1.2) admits a unique solution u(x) ∈ H2(R2) by means of Lemma 4 below along with Theorem 1. Let us apply the standard
Fourier transform (4.1) to both sides of (1.2) and (2.7). This yields
G(p)ϕ(p) u(p) = 2π p2 + |p2|2s ,
1
um(p) = 2π Gm(p)ϕm(p) , p21 + |p2|2s
(3.5)
p2u(p) = 2π p2G(p)ϕ(p) , p21 + |p2|2s
p2u (p) = 2π p2Gm(p)ϕm(p) ,
m
p21 + |p2|2s
m ∈ N,
(3.6)
where ϕ(p) and ϕm(p) stand for the Fourier images of F (u(x), x) and F (um(x), x) respec-
tively. Apparently,
|um(p) − u(p)| ≤ 2π Gm(p) − G(p)
|ϕ(p)|+
p21 + |p2|2s p21 + |p2|2s L∞(R2)
Hence
+2π Gm(p)
|ϕm(p) − ϕ(p)|.
p21 + |p2|2s L∞(R2)
um − u L2(R2) ≤ 2π Gm(p) − G(p)
F (u(x), x) + L2(R2)
p21 + |p2|2s p21 + |p2|2s L∞(R2)
+2π Gm(p)
F (um(x), x) − F (u(x), x) . L2(R2)
p21 + |p2|2s L∞(R2)
Upper bound (2.2) of Assumption 1 gives us
F (um(x), x) − F (u(x), x) L2(R2) ≤ l um(x) − u(x) . L2(R2)
(3.7)
Note that um(x), u(x) ∈ H2(R2) ⊂ L∞(R2) due to the Sobolev embedding. Thus, we arrive
at
um(x) − u(x) L2(R2) 1 − 2π Gm(p)
l≤
p21 + |p2|2s L∞(R2)
≤ 2π Using (4.24), we derive
Gm(p) − G(p) p21 + |p2|2s p21 + |p2|2s
L∞ (R2 )
F (u(x), x) . L2(R2)
um(x) − u(x) L2(R2) ≤ 2π Gm(p) − G(p)
F (u(x), x) . L2(R2)
ε p21 + |p2|2s p21 + |p2|2s L∞(R2)
8
Upper bound (2.1) of Assumption 1 gives us F (u(x), x) ∈ L2(R2) for u(x) ∈ L2(R2). Hence, we obtain that under the given conditions
um(x) → u(x), m → ∞
(3.8)
in L2(R2) due to the result of Lemma 4 of the Appendix. By virtue of (3.6), we arrive at
|p2u (p) − p2u(p)| ≤ 2π p2Gm(p) − p2G(p)
|ϕ(p)|+
m
p21 + |p2|2s p21 + |p2|2s L∞(R2)
Therefore,
p2Gm(p)
+2π p2 + |p2|2s
|ϕm(p) − ϕ(p)|.
1
L∞ (R2 )
p2Gm(p)
p2G(p)
∆um(x) − ∆u(x) L2(R2) ≤ 2π p2 + |p2|2s − p2 + |p2|2s
F (u(x), x) + L2(R2)
1
1
L∞ (R2 )
p2Gm(p)
+2π p2 + |p2|2s
F (um(x), x) − F (u(x), x) . L2(R2)
1
L∞ (R2 )
Inequality (3.7) enables us to obtain the upper bound
p2Gm(p)
p2G(p)
∆um(x) − ∆u(x) L2(R2) ≤ 2π p2 + |p2|2s − p2 + |p2|2s
F (u(x), x) + L2(R2)
1
1
L∞ (R2 )
p2Gm(p)
+2π p2 + |p2|2s
l um(x) − u(x) . L2(R2)
1
L∞ (R2 )
By means of the result of Lemma 4 of the Appendix along with (3.8), we derive ∆um(x) → ∆u(x) in L2(R2) as m → ∞. Definition (2.5) of the norm gives us um(x) → u(x) in H2(R2)
as m → ∞.
Suppose the solution um(x) of problem (2.7) studied above vanishes in the space of two dimensions for a certain m ∈ N. This will contradict to the given condition that the Fourier transforms of Gm(x) and F (0, x) are nontrivial on a set of nonzero Lebesgue measure in R2. The analogous argument is valid for the solution u(x) of limiting equation (1.2).
4 Appendix
Let G(x) be a function, G(x) : R2 → R, for which we denote its standard Fourier transform using the hat symbol as
G(p) := 1 G(x)e−ipxdx, 2π R2
p ∈ R2,
(4.1)
9
such that and G(x) = 1
2π quantities
1 G(p) L∞(R2) ≤ 2π G L1(R2)
(4.2)
G(q)eiqxdq, x ∈ R2. For the technical purposes we introduce the auxiliary
R2
N2, s := max
G(p) p21 + |p2|2s L∞(R2),
p2G(p) p21 + |p2|2s L∞(R2) ,
0 < s < 1.
(4.3)
Lemma 3. Let 0 < s < 1, the function G(x) : R2 → R, such that G(x) ∈ L1(R2) and x2G(x) ∈ L1(R2). We also assume that (−∆)1−sG(x) ∈ L1(R2).
1 a) If 0 < s ≤ 2 then N2, s < ∞ if and only if
(G(x), 1)L2(R2) = 0,
(4.4)
(G(x), x1)L2(R2) = 0.
(4.5)
b) Suppose 1 < s < 1. 2
(4.5) along with
Then N2, s < ∞ if and only if orthogonality conditions (4.4) and
(G(x), x2)L2(R2) = 0
(4.6)
hold.
Proof. Let us first observe that in both cases a) and b) of our lemma the boundedness of
G(p)
p2G(p)
p2 + |p2|2s yields that p2 + |p2|2s is bounded as well. We easily express
1
1
p2G(p)
p2G(p)
p2G(p)
p2 + |p2|2s = p2 + |p2|2s χ{|p|≤1} + p2 + |p2|2s χ{|p|>1}.
1
1
1
(4.7)
Here and further down χA will denote the characteristic function of a set A ⊆ R2. Clearly, the first term in the right side of (4.7) can be estimated from above in the absolute value by
G(p)
p2 + |p2|2s
<∞
1
L∞ (R2 )
as assumed. Inequality (4.2) gives us
|p|2(1−s)G(p) L∞(R2) ≤ 1 (−∆)1−sG(x) L1(R2). 2π
(4.8)
The right side of (4.8) is finite due to the one of our assumptions. In the polar coordinates
we have p = (|p|cosθ, |p|sinθ) ∈ R2,
10
Messoud Efendiev1,2, Vitali Vougalter3 1 Helmholtz Zentrum Mu¨nchen, Institut fu¨r Computational Biology, Ingolsta¨dter Landstrasse 1
Neuherberg, 85764, Germany e-mail: [email protected] 2 Department of Mathematics, Marmara University, Istanbul, Turkey
e-mail: [email protected]
3 Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada e-mail: [email protected]
Abstract. We establish the existence in the sense of sequences of solutions for certain integro-differential type equations in two dimensions involving the normal diffusion in one direction and the anomalous diffusion in the other direction in H2(R2) via the fixed point technique. The elliptic equation contains a second order differential operator without the Fredholm property. It is proved that, under the reasonable technical conditions, the convergence in L1(R2) of the integral kernels implies the existence and convergence in H2(R2) of the solutions.
Keywords: solvability conditions, non Fredholm operators, integro-differential equations, mixed diffusion AMS subject classification: 35J05, 35P30, 47F05
1 Introduction
Let us recall that a linear operator L acting from a Banach space E into another Banach space F satisfies the Fredholm property if its image is closed, the dimension of its kernel and the codimension of its image are finite. As a consequence, the equation Lu = f is solvable if and only if φi(f ) = 0 for a finite number of functionals φi from the dual space F ∗. These properties of Fredholm operators are widely used in many methods of linear and nonlinear analysis.
Elliptic problems in bounded domains with a sufficiently smooth boundary satisfy the Fredholm property if the ellipticity condition, proper ellipticity and Shapiro-Lopatinskii conditions are fulfilled (see e.g. [1], [6], [16], [20]). This is the main result of the theory of linear
1
elliptic equations. In the case of unbounded domains, these conditions may not be sufficient and the Fredholm property may not be satisfied. For instance, the Laplace operator, Lu = ∆u, in Rd does not satisfy the Fredholm property when considered in H¨older spaces, L : C2+α(Rd) → Cα(Rd), or in Sobolev spaces, L : H2(Rd) → L2(Rd).
Linear elliptic problems in unbounded domains satisfy the Fredholm property if and only if, in addition to the conditions cited above, limiting operators are invertible (see [21]). In some simple cases, limiting operators can be explicitly constructed. For instance, if
Lu = a(x)u′′ + b(x)u′ + c(x)u, x ∈ R,
where the coefficients of the operator have limits at infinity,
a± = lim a(x), x→±∞
the limiting operators are:
b± = lim b(x), x→±∞
c± = lim c(x), x→±∞
L±u = a±u′′ + b±u′ + c±u.
Since the coefficients are constants, the essential spectrum of the operator, that is the set of complex numbers λ for which the operator L − λ fails to satisfy the Fredholm property, can be explicitly found by virtue of the Fourier transform:
λ±(ξ) = −a±ξ2 + b±iξ + c±, ξ ∈ R.
Invertibility of limiting operators is equivalent to the condition that the essential spectrum
does not contain the origin.
In the case of general elliptic problems, the same assertions hold true. The Fredholm
property is satisfied if the essential spectrum does not contain the origin or if the limiting
operators are invertible. However, these conditions may not be explicitly written.
In the case of non-Fredholm operators the usual solvability conditions may not be ap-
plicable and solvability conditions are, in general, not known. There are some classes of
operators for which solvability conditions are derived. We illustrate them with the following
example. Consider the problem
Lu ≡ ∆u + au = f
(1.1)
in Rd, where a is a positive constant. The operator L coincides with its limiting operators.
The homogeneous equation has a nonzero bounded solution. Hence the Fredholm property is
not satisfied. However, since the operator has constant coefficients, we can apply the Fourier
transform and find the solution explicitly. Solvability relations can be formulated as follows.
If f ∈ L2(Rd) and xf ∈ L1(Rd), then there exists a solution of this problem in H2(Rd) if
and only if
f (x), eipx
= 0, p ∈ S√d a.e.
d
(2π) 2 L2(Rd)
a
(see [26]). Here and further down Srd stands for the sphere in Rd of radius r centered at the origin. Thus, though the operator does not satisfy the Fredholm property, solvability
2
relations are formulated similarly. However, this similarity is only formal since the range of the operator is not closed.
In the case of the operator with a potential,
Lu ≡ ∆u + a(x)u = f,
Fourier transform is not directly applicable. Nevertheless, solvability relations in R3 can be derived by a rather sophisticated application of the theory of self-adjoint operators (see [24]). As before, solvability conditions are formulated in terms of orthogonality to solutions of the homogeneous adjoint equation. There are several other examples of linear elliptic non Fredholm operators for which solvability conditions can be obtained (see [11], [21], [22], [23], [24], [26]).
Solvability relations play a crucial role in the analysis of nonlinear elliptic problems. In the case of non-Fredholm operators, in spite of some progress in understanding of linear equations, there exist only few examples where nonlinear non-Fredholm operators are analyzed (see [5], [9], [25], [26], [29]). The large time behavior of solutions of a class of fourth-order parabolic equations defined on unbounded domains using the Kolmogorov ε-entropy as a measure was studied in [8]. The work [7] deals with the finite and infinite dimensional attractors for evolution equations of mathematical physics. The attractor for a nonlinear reaction-diffusion system in an unbounded domain in R3 was investigated in [12]. The articles [13] and [19] are devoted to the understanding of the Fredholm and properness properties of quasilinear elliptic systems of second order and of operators of this kind on RN . Exponential decay and Fredholm properties in second-order quasilinear elliptic systems were addressed in [14]. In the present article we treat another class of stationary nonlinear problems, for which the Fredholm property may not be satisfied:
∂2u ∂x2 −
1
∂2 s
− ∂x2 u + G(x − y)F (u(y), y)dy = 0,
2
R2
0 < s < 1,
(1.2)
where x = (x1, x2) ∈ R2, y = (y1, y2) ∈ R2. Here the operator
∂2 Ls := − ∂x2 +
1
∂2 − ∂x2
2
s
: H2(R2) → L2(R2),
0
(1.3)
is defined via the spectral calculus. The novelty of the present work is that in the diffusion term we add the standard minus Laplacian in the x1 variable with the negative Laplace operator in x2 raised to a fractional power. Such model is new and not much is understood about it, especially in the context of the integro-differential equations. The difficulty we have to overcome is that such problem becomes anisotropic and it is more technical to obtain the desired estimates when dealing with it. In population dynamics in the Mathematical Biology the integro-differential problems describe models with intra-specific competition and nonlocal consumption of resources (see e.g. [2], [3]). It is important to study the equations of this kind in unbounded domains from the point of view of the understanding of the spread of the viral infections, since many countries have to deal with the pandemics. We use the explicit
3
form of the solvability conditions and establish the existence of solutions of such nonlinear equation. In the case of the standard Laplacian instead of (1.3), the problem analogous to (1.2) was considered in [25] and [29]. The solvability of the integro-differential equations involving in the diffusion term only the negative Laplace operator raised to a fractional power was actively studied in recent years in the context of the anomalous diffusion (see e.g. [10], [27], [28]). The anomalous diffusion can be described as a random process of particle motion characterized by the probability density distribution of jump length. The moments of this density distribution are finite in the case of the normal diffusion, but this is not the case for the anomalous diffusion. The asymptotic behavior at infinity of the probability density function determines the value of the power of the Laplacian (see [18]). In article [17] the authors prove the imbedding theorems and study the spectrum of certain pseudodifferential operators.
2 Formulation of the results
The nonlinear part of equation (1.2) will satisfy the following regularity conditions.
Assumption 1. Function F (u, x) : R × R2 → R is satisfying the Caratheodory condition (see [15]), such that
|F (u, x)| ≤ k|u| + h(x) f or u ∈ R, x ∈ R2
(2.1)
with a constant k > 0 and h(x) : R2 → R+, h(x) ∈ L2(R2). Moreover, it is a Lipschitz continuous function, such that
|F (u1, x) − F (u2, x)| ≤ l|u1 − u2| f or any u1,2 ∈ R, x ∈ R2
(2.2)
with a constant l > 0.
The solvability of a local elliptic equation in a bounded domain in RN was considered in [4], where the nonlinear function was allowed to have a sublinear growth. In order to study of the existence of solutions of (1.2), we introduce the auxiliary equation
∂2u − ∂x2 +
1
∂2 s
− ∂x2 u = G(x − y)F (v(y), y)dy,
2
R2
0 < s < 1.
(2.3)
We denote
(f1(x), f2(x))L2(R2) := f1(x)f¯2(x)dx,
R2
(2.4)
with a slight abuse of notations when these functions are not square integrable. Indeed,
if f1(x) ∈ L1(R2) and f2(x) is bounded, like for instance those involved in orthogonality
relation (4.4) below, the integral in the right side of (2.4) makes sense. In the the article we
work in the space of the two dimensions, such that the appropriate Sobolev space is equipped
with the norm
u
2 H 2 (R2 )
:=
u
2 L2 (R2 )
+
∆u
. 2
L2 (R2 )
(2.5)
4
In the equation above we are dealing with the operator Ls defined in (1.3). By virtue of
the standard Fourier transform (4.1), it can be easily checked that its essential spectrum is
given by
λs(p) = p21 + |p2|2s, p = (p1, p2) ∈ R2.
(2.6)
Since set (2.6) contains the origin, our operator Ls fails to satisfy the Fredholm property, which is the obstacle to solve our equation.
The similar situations but in linear problems, both self- adjoint and non self-adjoint involving non Fredholm differential operators have been studied extensively in recent years (see [21], [22], [24], [26]). Our present work is related to our article [11] since we also deal with the non Fredholm operator, now involved in the problem, which is not linear anymore and contains the nonlocal terms. Currently, as distinct from [11], the space dimension is restricted to d = 2 to avoid the extra technicalities.
In the present work we manage to establish that under the reasonable technical assumptions problem (2.3) defines a map T2, s : H2(R2) → H2(R2), 0 < s < 1, which is a strict contraction.
Theorem 1. Let Assumption 1 hold, 0 < s < 1, the function G(x) : R2 → R, such that G(x) ∈ L1(R2) and x2G(x) ∈ L1(R2). Moreover, (−∆)1−sG(x) ∈ L1(R2).
We also assume that orthogonality conditions (4.4), (4.5) hold if 0 < s ≤ 1 and relations
1
√
2
(4.4), (4.5) and (4.6) are valid for 2 < s < 1 and that 2 2πN2, sl < 1. Then the map
T2,sv = u on H2(R2) defined by problem (2.3) admits a unique fixed point v2,s, which is the
only solution of equation (1.2) in H2(R2).
This fixed point v2,s is nontrivial provided the intersection of supports of the Fourier transforms of functions suppF (0, x) ∩ suppG is a set of nonzero Lebesgue measure in R2.
Related to equation (1.2) in the space of two dimensions, we study the sequence of approximate equations with m ∈ N
∂2um ∂x2 −
1
∂2 s
− ∂x2 um + Gm(x − y)F (um(y), y)dy = 0,
2
R2
0 < s < 1.
(2.7)
The sequence of kernels {Gm(x)}∞ m=1 tends to G(x) as m → ∞ in the appropriate function spaces discussed below. We will show that, under the appropriate technical conditions, each of equations (2.7) has a unique solution um(x) ∈ H2(R2), the limiting problem (1.2) admits a unique solution u(x) ∈ H2(R2), and um(x) → u(x) in H2(R2) as m → ∞, which is the so-called existence of solutions in the sense of sequences. In this case, the solvability relations
can be formulated for the iterated kernels Gm. They yield the convergence of the kernels in terms of the Fourier transforms (see the Appendix) and, as a consequence, the convergence
or the solutions (Theorem 2 below). Similar ideas in the context of the standard Schro¨dinger
type operators were exploited in [23]. Our second main proposition is as follows.
5
Theorem 2. Let Assumption 1 hold, 0 < s < 1, m ∈ N, the functions Gm(x) : R2 → R are such that Gm(x) ∈ L1(R2), x2Gm(x) ∈ L1(R2) and (−∆)1−sGm(x) ∈ L1(R2). Moreover, Gm(x) → G(x) in L1(R2), x2Gm(x) → x2G(x) and (−∆)1−sGm(x) → (−∆)1−sG(x) in L1(R2) as m → ∞. We also assume that for all m ∈ N orthogonality conditions (4.21), (4.22) hold if 0 < s ≤ 1
12 and relations (4.21), (4.22) and (4.23) are valid for 2 < s < 1. Furthermore, we suppose that (4.24) holds for all m ∈ N with a certain 0 < ε < 1.
Then each problem (2.7) admits a unique solution um(x) ∈ H2(R2), limiting equation (1.2) possesses a unique solution u(x) ∈ H2(R2) and um(x) → u(x) in H2(R2) as m → ∞.
The unique solution um(x) of each problem (2.7) is nontrivial provided that the intersection of supports of the Fourier transforms of functions suppF (0, x) ∩ suppGm is a set of nonzero Lebesgue measure in R2. Similarly, the unique solution u(x) of limiting equation (1.2) does not vanish identically if suppF (0, x) ∩ suppG is a set of nonzero Lebesgue measure in R2.
Remark 1. In the article we work with real valued functions by virtue of the assumptions on F (u, x), Gm(x) and G(x) involved in the nonlocal terms of the iterated and limiting problems discussed above.
Remark 2. The importance of Theorem 2 above is the continuous dependence of solutions with respect to the integral kernels.
3 Proofs Of The Main Results
Proof of Theorem 1. Let us first suppose that for a certain v(x) ∈ H2(R2) there exist two
solutions u1,2(x) ∈ H2(R2) of problem (2.3). Then their difference w(x) := u1(x) − u2(x) ∈ H2(R2) will be a solution of the homogeneous equation
∂2w − ∂x2 +
1
∂2 − ∂x2
2
s
w = 0.
Because the operator Ls : H2(R2) → L2(R2) defined in (1.3) does not have any nontrivial zero modes, the function w(x) vanishes in the space of two dimensions.
We choose arbitrarily v(x) ∈ H2(R2). Let us apply the standard Fourier transform (4.1)
to both sides of (2.3) and arrive at
G(p)f (p) u(p) = 2π p2 + |p2|2s ,
1
p2u(p) = 2π p2G(p)f (p), p21 + |p2|2s
(3.1)
where f (p) denotes the Fourier image of F (v(x), x). Evidently, we have the estimates from
above |u(p)| ≤ 2πN2, s|f (p)| and |p2u(p)| ≤ 2πN2, s|f (p)|.
6
Note that N2, s < ∞ by means of Lemma 3 of the Appendix under the given conditions. This enables us to obtain the upper bound on the norm
u
2 H 2 (R2 )
=
u(p)
2 L2 (R2 )
+
p2u(p)
2 L2 (R2 )
≤
8π2N22,
s
F (v(x), x)
, 2
L2 (R2 )
which is finite by virtue of (2.1) of Assumption 1 because v(x) ∈ L2(R2). Clearly, v(x) ∈ H2(R2) ⊂ L∞(R2) due to the Sobolev embedding. Thus, for an arbitrary v(x) ∈ H2(R2) there exists a unique solution u(x) ∈ H2(R2) of problem (2.3), such that its Fourier image is given by (3.1). Hence, the map T2,s : H2(R2) → H2(R2) is well defined. This allows us to choose arbitrary functions v1,2(x) ∈ H2(R2), such that their images u1,2 := T2,sv1,2 ∈ H2(R2).
Clearly, (2.3) yields
∂2u1 − ∂x2 +
1
∂2 s
− ∂x2 u1 = G(x − y)F (v1(y), y)dy,
2
R2
(3.2)
− ∂2u2 + ∂x21
∂2 s
− ∂x2 u2 = G(x − y)F (v2(y), y)dy,
2
R2
(3.3)
where 0 < s < 1. Let us apply the standard Fourier transform (4.1) to both sides of the equations of system (3.2), (3.3) above. We arrive at
u1(p) = 2π G(p)f1(p) , p21 + |p2|2s
u2(p) = 2π G(p)f2(p) . p21 + |p2|2s
(3.4)
Here f1(p) and f2(p) stand for the Fourier images of F (v1(x), x) and F (v2(x), x) respectively. By means of (3.4) we derive the upper bounds
|u1(p) − u2(p)| ≤ 2πN2, s|f1(p) − f2(p)|, p2|u1(p) − u2(p)| ≤ 2πN2, s|f1(p) − f2(p)|,
such that
u1 − u2
2 H 2 (R2 )
=
u1(p) − u2(p)
2 L2 (R2 )
+
p2[u1(p) − u2(p)]
2 L2 (R2 )
≤
≤ 8π2N22, s
F (v1(x), x) − F (v2(x), x)
. 2
L2 (R2 )
Evidently, v1,2(x) ∈ H2(R2) ⊂ L∞(R2) via the Sobolev embedding. Condition (2.2) above
implies that
√ T2,sv1 − T2,sv2 H2(R2) ≤ 2 2πN2, sl v1 − v2 H2(R2)
and the constant in the right side of this inequality is less than one via the one of our
assumptions. Thus, by means of the Fixed Point Theorem, there exists a unique function v2,s ∈ H2(R2) with the property T2,sv2,s = v2,s, which is the only solution of problem (1.2) in H2(R2). Suppose v2,s(x) = 0 identically in the space of two dimensions. This will contradict to our assumption that the Fourier images of G(x) and F (0, x) do not vanish on a set of nonzero Lebesgue measure in R2.
7
Let us proceed to establishing the solvability in the sense of sequences for our integrodifferential problem in the space of two dimensions.
Proof of Theorem 2. By virtue of the result of Theorem 1 above, each problem (2.7) has a unique solution um(x) ∈ H2(R2), m ∈ N. Limiting equation (1.2) admits a unique solution u(x) ∈ H2(R2) by means of Lemma 4 below along with Theorem 1. Let us apply the standard
Fourier transform (4.1) to both sides of (1.2) and (2.7). This yields
G(p)ϕ(p) u(p) = 2π p2 + |p2|2s ,
1
um(p) = 2π Gm(p)ϕm(p) , p21 + |p2|2s
(3.5)
p2u(p) = 2π p2G(p)ϕ(p) , p21 + |p2|2s
p2u (p) = 2π p2Gm(p)ϕm(p) ,
m
p21 + |p2|2s
m ∈ N,
(3.6)
where ϕ(p) and ϕm(p) stand for the Fourier images of F (u(x), x) and F (um(x), x) respec-
tively. Apparently,
|um(p) − u(p)| ≤ 2π Gm(p) − G(p)
|ϕ(p)|+
p21 + |p2|2s p21 + |p2|2s L∞(R2)
Hence
+2π Gm(p)
|ϕm(p) − ϕ(p)|.
p21 + |p2|2s L∞(R2)
um − u L2(R2) ≤ 2π Gm(p) − G(p)
F (u(x), x) + L2(R2)
p21 + |p2|2s p21 + |p2|2s L∞(R2)
+2π Gm(p)
F (um(x), x) − F (u(x), x) . L2(R2)
p21 + |p2|2s L∞(R2)
Upper bound (2.2) of Assumption 1 gives us
F (um(x), x) − F (u(x), x) L2(R2) ≤ l um(x) − u(x) . L2(R2)
(3.7)
Note that um(x), u(x) ∈ H2(R2) ⊂ L∞(R2) due to the Sobolev embedding. Thus, we arrive
at
um(x) − u(x) L2(R2) 1 − 2π Gm(p)
l≤
p21 + |p2|2s L∞(R2)
≤ 2π Using (4.24), we derive
Gm(p) − G(p) p21 + |p2|2s p21 + |p2|2s
L∞ (R2 )
F (u(x), x) . L2(R2)
um(x) − u(x) L2(R2) ≤ 2π Gm(p) − G(p)
F (u(x), x) . L2(R2)
ε p21 + |p2|2s p21 + |p2|2s L∞(R2)
8
Upper bound (2.1) of Assumption 1 gives us F (u(x), x) ∈ L2(R2) for u(x) ∈ L2(R2). Hence, we obtain that under the given conditions
um(x) → u(x), m → ∞
(3.8)
in L2(R2) due to the result of Lemma 4 of the Appendix. By virtue of (3.6), we arrive at
|p2u (p) − p2u(p)| ≤ 2π p2Gm(p) − p2G(p)
|ϕ(p)|+
m
p21 + |p2|2s p21 + |p2|2s L∞(R2)
Therefore,
p2Gm(p)
+2π p2 + |p2|2s
|ϕm(p) − ϕ(p)|.
1
L∞ (R2 )
p2Gm(p)
p2G(p)
∆um(x) − ∆u(x) L2(R2) ≤ 2π p2 + |p2|2s − p2 + |p2|2s
F (u(x), x) + L2(R2)
1
1
L∞ (R2 )
p2Gm(p)
+2π p2 + |p2|2s
F (um(x), x) − F (u(x), x) . L2(R2)
1
L∞ (R2 )
Inequality (3.7) enables us to obtain the upper bound
p2Gm(p)
p2G(p)
∆um(x) − ∆u(x) L2(R2) ≤ 2π p2 + |p2|2s − p2 + |p2|2s
F (u(x), x) + L2(R2)
1
1
L∞ (R2 )
p2Gm(p)
+2π p2 + |p2|2s
l um(x) − u(x) . L2(R2)
1
L∞ (R2 )
By means of the result of Lemma 4 of the Appendix along with (3.8), we derive ∆um(x) → ∆u(x) in L2(R2) as m → ∞. Definition (2.5) of the norm gives us um(x) → u(x) in H2(R2)
as m → ∞.
Suppose the solution um(x) of problem (2.7) studied above vanishes in the space of two dimensions for a certain m ∈ N. This will contradict to the given condition that the Fourier transforms of Gm(x) and F (0, x) are nontrivial on a set of nonzero Lebesgue measure in R2. The analogous argument is valid for the solution u(x) of limiting equation (1.2).
4 Appendix
Let G(x) be a function, G(x) : R2 → R, for which we denote its standard Fourier transform using the hat symbol as
G(p) := 1 G(x)e−ipxdx, 2π R2
p ∈ R2,
(4.1)
9
such that and G(x) = 1
2π quantities
1 G(p) L∞(R2) ≤ 2π G L1(R2)
(4.2)
G(q)eiqxdq, x ∈ R2. For the technical purposes we introduce the auxiliary
R2
N2, s := max
G(p) p21 + |p2|2s L∞(R2),
p2G(p) p21 + |p2|2s L∞(R2) ,
0 < s < 1.
(4.3)
Lemma 3. Let 0 < s < 1, the function G(x) : R2 → R, such that G(x) ∈ L1(R2) and x2G(x) ∈ L1(R2). We also assume that (−∆)1−sG(x) ∈ L1(R2).
1 a) If 0 < s ≤ 2 then N2, s < ∞ if and only if
(G(x), 1)L2(R2) = 0,
(4.4)
(G(x), x1)L2(R2) = 0.
(4.5)
b) Suppose 1 < s < 1. 2
(4.5) along with
Then N2, s < ∞ if and only if orthogonality conditions (4.4) and
(G(x), x2)L2(R2) = 0
(4.6)
hold.
Proof. Let us first observe that in both cases a) and b) of our lemma the boundedness of
G(p)
p2G(p)
p2 + |p2|2s yields that p2 + |p2|2s is bounded as well. We easily express
1
1
p2G(p)
p2G(p)
p2G(p)
p2 + |p2|2s = p2 + |p2|2s χ{|p|≤1} + p2 + |p2|2s χ{|p|>1}.
1
1
1
(4.7)
Here and further down χA will denote the characteristic function of a set A ⊆ R2. Clearly, the first term in the right side of (4.7) can be estimated from above in the absolute value by
G(p)
p2 + |p2|2s
<∞
1
L∞ (R2 )
as assumed. Inequality (4.2) gives us
|p|2(1−s)G(p) L∞(R2) ≤ 1 (−∆)1−sG(x) L1(R2). 2π
(4.8)
The right side of (4.8) is finite due to the one of our assumptions. In the polar coordinates
we have p = (|p|cosθ, |p|sinθ) ∈ R2,
10