Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space

We consider the local existence and the uniqueness of a weak solution of the initial boundary value problem to a convection–diffusion equation in a uniformly local function space \begin{document}$ L^r_{{\rm uloc}, \rho}( \Omega) $\end{document} , where the solution is not decaying at \begin{document}$ |x|\to \infty $\end{document} . We show that the local existence and the uniqueness of a solution for the initial data in uniformly local \begin{document}$ L^r $\end{document} spaces and identify the Fujita-Weissler critical exponent for the local well-posedness found by Escobedo-Zuazua [ 10 ] is also valid for the uniformly local function class.

1. Introduction. We consider the Cauchy problem of a convection-diffusion equation in spatially non-decaying function space. Let R n be the n-dimensional Euclidean space with n ≥ 1, and Ω ⊂ R n be an unbounded domain with uniform C 2 boundary. We consider the Cauchy-Dirichlet problem of a time dependent convection-diffusion equation: For a ∈ R n , where u = u(t, x); R + × Ω → R is the unknown function, and u 0 = u 0 (x); Ω → R is a given initial data.
For the case Ω = R n , Escobedo and Zuazua [10] showed that for initial data u 0 ∈ L 1 (R n ), there exists a unique global strong solution u ∈ C([0, ∞); L 1 (R n )) of (1.1) in u ∈ C((0, ∞); W 2,q (R n )) ∩ C 1 ((0, ∞); L q (R n )), (1.2) for every q ∈ (1, ∞). They also described the large time behavior of solutions of (1.1) and showed a decay property when the initial data is in L 1 (R n ). When p = 1 + 1 n , they proved that the large time behavior of solutions with initial data in L 1 (R n ) is given by a one-parameter family of self-similar solutions. The relevant parameter is the mass of the solution that is conserved for all t. When p > 1 + 1 n , the convection term is too weak and they proved that the large time behavior of solutions is given by the heat kernel.
On the other hand, in [3,4,7,19,20,22] and [23], the authors make use of spaces of functions which have the property that their elements have some uniform size when it is measured in balls of fixed radius but arbitrary center. These spaces are called as uniformly local spaces. These spaces are natural and useful for finding the solutions of parabolic equations in unbounded domains with non-decaying initial functions. The spaces enjoy suitable inclusion properties and have locally compact embeddings and besides any constant functions belong to them. In particular, when we analyze parabolic equations in unbounded domains, these spaces will allow us to consider a solution with no prescribed behavior at infinity and allowing for local singularities.  Here we identify L ∞ uloc,ρ (Ω) as L ∞ (Ω). The space L r uloc,ρ (Ω) is a Banach space with the norm defined in (1.3). We define the subspace L r uloc,ρ (Ω) as the closure of the space of bounded uniformly continuous functions BU C(Ω) in the space L r uloc,ρ (Ω), i.e., L r uloc,ρ (Ω) := BU C(Ω) · L r uloc,ρ and define L ∞ uloc,ρ (Ω) = BU C(Ω). The Sobolev spaces W k,r uloc,ρ (Ω) for 1 ≤ r ≤ ∞, ρ > 0 and k = 1, 2, . . . are analogously introduced. We define by
Definition 1.2 (Weak L r uloc (Ω)-solution). Let 1 ≤ r < ∞ and ρ > 0. For an initial data u 0 ∈ L r uloc,ρ (Ω) and T > 0, we say that u is a weak L r uloc (Ω)-solution of (1.1) We state our main result for the existence of a weak solution to (1.1) in uniformly local L r spaces. Theorem 1.3 (Existence of a weak solution). Let p > 1 and 1 ≤ r < ∞ with (1.5) There exists a positive constant γ 0 , depending only on n, p and r, such that, if for any initial data u 0 ∈ L r uloc,ρ (Ω) satisfies for some ρ > 0, then there exists a unique weak L r where C is independent of u. Besides the solution has a uniform estimate and hence u ∈ L ∞ (0, ρ 2 ) × Ω .
In the assumption on the initial data (1.6), the constant γ 0 > 0 is a constant depending only on n, p and r. Hence one can regard this condition on the initial data as the restriction on the choice of ρ > 0. Since the function class L r uloc,ρ (Ω) does not depend on ρ > 0, we have a room for the choice of ρ > 0 depending on the initial data. This choice is reflecting how long the local solution can be continued.
As a corollary of Theorem 1.3, we have: Then there exists a constant γ such that, if u 0 ∈ L n(p−1) (Ω) and u 0 L n(p−1) (Ω) ≤ γ, then problem (1.1) has a global solution.
The local well-posedness problem for the Fujita type nonlinear heat equation was discussed by many authors: For 1 < p < ∞ and a = 0, In particular, Weissler [26] obtained the sharp well-posedness result in the Lebesgue where u λ also solves the equation (1.9). The threshold scaling space appears when the exponent of the coefficient λ 2 p−1 of the scaled function (1.10) coincides the L 1 invariant scaling. The corresponding result to the convection-diffusion equation (1.1) also holds for the critical exponent p = 1 + 1 n (cf. [10]). In our case, even in the uniformly local spaces, the well-posedness threshold coincides with the usual Lebesgue space case. This stands for that the role of exponent of the function space essentially limited in a local sense and the behavior of the solution at spacial infinity does not give a large difference for the time local well-posedness as far as the function space remains in a uniform sense.
Note that since the above weak solution does not decay at |x| → ∞, it does not satisfy the L 1 conservation law anymore. Hence the global existence of the solution is not clear.

2.
Preliminaries. In this section, we review some fundamental inequalities which will be used throughout this paper. Lemma 2.1. Let u ∈ W 1,p (Ω), then u + , u − , |u| ∈ W 1,p (Ω) and Besides it holds that

CONVECTION-DIFFUSION EQUATIONS IN UNIFORMLY LOCAL LEBESGUE SPACES 681
For the proof see [17]. Here we summarize important properties for functions belonging to the uniformly local L r spaces.
for some constant C depending only on n, ρ and ρ if ρ > ρ.
Remark 1. The class of compact supported smooth functions; C ∞ 0 (Ω) is not dense in L r uloc,ρ (Ω). Proof. Proposition 2.1 (1) The inclusion follows from the Hölder inequality. (2) For the case ρ ≤ ρ, the inequality (2.11) directly follows from the definition of the uniform local norm with the constant C = 1. Hence we assume ρ > ρ. Let Since N is only depending on n, it yields that Then there exists a constant C > 0 such that for any function (2.12) Lemma 2.3 (The Gagliardo-Nirenberg inequality). Let 0 < r ≤ ∞, 1 ≤ p, q ≤ ∞, and θ ∈ [0, 1] satisfying

MD. RABIUL HAQUE, TAKAYOSHI OGAWA AND RYUICHI SATO
Then there exists a constant C GN > 0, depending only on p, q, r and n such that For the proof see ( [15], [24]).
Proposition 2.4 (Existence of a strong solution). Let n ≥ 1. Then for any u 0 ∈ BU C(Ω) with u 0 = 0 on ∂Ω, there exist T = T ( u 0 L ∞ (Ω) ) > 0 and a unique mild solution u ∈ C [0, T ); BU C(Ω) to (1.1), i.e., Proof. Proposition 2.4 For M > 0 and T > 0, we let where M and T are constants depending on u 0 L ∞ (Ω) and p and n determined later. X T is a complete metric space with the metric; for any f and g ∈ X T ,

CONVECTION-DIFFUSION EQUATIONS IN UNIFORMLY LOCAL LEBESGUE SPACES 683
Then we consider a map and show that Φ is a contraction mapping from X T to itself. This implies an existence of the fixed point for the map Φ on X T and it becomes a solution to the corresponding integral equation (2.17) has a unique fixed point and it becomes an L r -mild solution.
The first term related to the initial data u 0 in the right hand side of (2.18) is directly estimated by the dissipative estimate as follows; While for the second part of the norm ||| · |||, Hence combining (2.19) and (2.20), we obtain We show for T > 0, the following two estimates to be verified: If T > 0 is sufficiently small depending on n, p and u 0 ∞ , then for any u and v ∈ X T .
Proof. Lemma 2.5 For some T > 0, let I = (0, T ) and M = 4 max(1, C α ) u 0 ∞ . To see the estimate (2.22) hold, we invoke the L p -L q type dissipative estimate  . Since the first term of the right hand side of (2.18) is common, we have for any u and v ∈ X T , that (2.29) Hence by setting Similar to the bound estimate, we proceed (2.32) Hence by (2.30), we conclude from (2.32) that

CONVECTION-DIFFUSION EQUATIONS IN UNIFORMLY LOCAL LEBESGUE SPACES 685
Now we see from Lemma 2.5 that the map Φ is contraction from X T to X T and by virtue of the Banach fixed point theorem, there exists a unique fixed point of Φ in X T . By the definition, this fixed point satisfies the integral equation (2.17) and besides, u(t) → u 0 as t → 0. Hence u is the L ∞ -mild solution to (1.1). This shows the existence of solution. The uniqueness and the continuous dependence of the initial data is obtained by very similar estimate (2.29).
Since the solution is in X T , it is bounded uniformly continuous in x variable for all t < T . Besides the solution is belonging to W α,∞ (Ω) and hence it is Hölder continuous in x. Then the mild solution satisfies the equation (1.1) with the initial data pointwisely and this completes the proof of Proposition 2.4.

3.
A priori estimates. In this section, we give some a priori estimates for a weak solution to (1.1). All the estimates holds for the weak solution to (1.1) if we assume that the solution exists. In what follows, we denote B ρ (x) ∩ Ω for x ∈ Ω, ρ > 0 by simply B ρ (x) unless otherwise specified. for 0 < t ≤ min{µρ 2 , T }, where C * is a positive constant depending only on n, p and r.
Proposition 3.2 (Difference estimate). Let r satisfy (1.5), r > 1 and T > 0. Let u 0 and v 0 ∈ L r uloc,ρ (Ω) be two initial data and suppose that u and v be a corresponding L r uloc (Ω)-solution of (1.1) in (0, T ) × Ω, respectively. Then there exists a positive constant γ 2 such that, if for some ρ > 0, then there exists µ > 0 depending only on p, r, n and γ 2 such that for 0 < t ≤ min{µρ 2 , T }, where C is a positive constants depending only on n, p and r.
Proof. Proposition 3.2 Let x ∈ Ω and ζ be a smooth function in C ∞ 0 (Ω) defined in (3.3). Suppose that u and v are two strong solutions of (1.1) in (0, T ) × Ω and let w = u − v. Then multiply |w| r−1 (s, y)(sgn w(s, y))ζ k (s, y) for k ∈ N to the difference of equation and integrate it over Ω, we obtain 1 r d dt Ω |w(s)| r ζ k dy + Ω ∇w(s) · ∇(|w(s)| r−1 (sgn w(s))ζ k dy Observing that By the mean value theorem (3.20)