Existence of weak solutions to a convection– diffusion equation in amalgam spaces

We consider the local existence and uniqueness of a weak solution for a convection– diffusion equation in amalgam spaces. We establish the local existence and uniqueness of solution for the initial condition in amalgam spaces. Furthermore, we prove the validity of the Fujita–Weissler critical exponent for local existence and uniqueness of solution in the amalgam function class that is identified by Escobedo and Zuazua (J Funct Anal 100:119–161, 1991).

spaces more applicable than to Lebesgue spaces and uniformly local Lebesgue spaces because the Lebesgue and uniformly local Lebesgue norm does not distinguish between local and global properties. Amalgam space has a long history and has been studied by many authors, [4-7, 16, 20, 25], etc. Amalgam spaces arise naturally in harmonic analysis. In 1926, Norbert Wiener, who was the first one to introduce the amalgam spaces, considers some special cases in [29][30][31]. Amalgams have been reinvented many times in the literature; the first systematic study appears by Holland in [23]; an excellent review article is [17]. H. Feichtinger [13][14][15] introduced a far-reaching generalization of amalgam spaces to general topological groups and general local/global function spaces.
To this end, we introduce the notion of weak solutions to (1.1) in amalgam spaces L r,ν ρ (�) as follows.
There exists a positive constant γ 0 , depending only on n, p and r, such that, if for any initial condition u 0 ∈ L r,ν ρ (�) satisfies for some ρ > 0 , then there exists a unique weak L r,ν (�) -solution u of (1.1) in (0, µρ 2 ) × � such that where C and µ are independent of u. Besides the solution has a uniform estimate and hence u ∈ L ∞ (0, µρ 2 ) × � for some µ > 0.
Local well-posedness problem for Fujita-type nonlinear heat equation was discussed by many authors: For 1 < p < ∞, In particular, Weissler [28] obtained the sharp well-posedness result in Lebesgue spaces: If then solution exists and well-posed in Lebesgue spaces L r (�) . The exponent appears naturally from the invariant scaling equipped with the equation itself; where u also solves equation (1.5). The threshold scaling space appears when the exponent of the coefficient 2 p−1 of the scaled function (1.6) coincides the L 1 invariant scaling. The corresponding result to the convection-diffusion equations (1.1) also holds for the critical exponent p = 1 + 1 n (cf. [9]). Our main finding is that even in amalgam spaces decouple the connection between local and global properties that is inherent in Lebesgue spaces, the well-posedness threshold coincides with the usual Lebesgue spaces case. Furthermore, amalgam spaces are a space between usual Lebesgue spaces and uniformly local Lebesgue spaces. We compare our result with the result of [21] and obtain stronger conclusion even though our initial data class smaller than that of [21]. This paper is organized as follows. In "Preliminaries" section, we will state some properties of amalgam spaces. In "A priori estimates" section, we will prove our key estimates: a priori estimates, difference estimates and L ∞ estimates for a weak solution in amalgam spaces. In "Proof of Theorem" section, we will prove our main Theorem 1.1 using the estimates that proved in "A priori estimates" section.

Preliminaries
In this section, we present important properties for functions belonging to amalgam spaces that will be used later.

Proposition 2.2 The class of compact-supported smooth functions
For the proof, see ( [25]).

Proposition 2.4
Let n ≥ 1 , ⊂ R n , x 0 ∈ , ρ > 0 and 1 ≤ p, q, r < ∞ with Then, there exists a constant C > 0 such that for any function f satisfying For the proof, see [21] A priori estimates In this section, we give some a priori estimates for a weak solution to (1.1). All the estimates hold for the weak solutions to (1.1) if we assume that the solutions exist. In the remainder of this paper, we denote B ρ (x) ∩ � for x ∈ , ρ > 0 by simply B ρ (x) unless otherwise specified.
for some ρ > 0 , then there exists a constant µ > 0 depending only on p, r, n and γ 1 such that for 0 < t < min{µρ 2 , T } , where C is a positive constant depending only on n, p and r.

Proof of Proposition 3.2
Let x ∈ and ζ be a smooth function in C ∞ 0 (�) defined in (1.1) Suppose that u and v are two strong solutions of (1.1) in (0, T ) × � and let w = u − v . Then multiply |w| r−1 (sgn w)ζ k for k ∈ N to the difference of equation and integrate it over we obtain that Observing that By mean value's theorem, we know that Therefore, by (3.14) and (3.15), we obtain from (3.13) that (3.14) ∇w · ∇(|w| r−1 (sgn w)ζ k ) ≥ C 1 ∇(|w| max(|u(s)|, |v(s)|) p−1 ∇|w(s)| r ζ k dy max(|u(s)|, |v(s)|) p−1 |w(s)| r |∇ζ k |dy. Now we estimate the first and last term of the right hand side of (3.16) using the Young and the Hölder inequalities. The first term of the right hand side of (3.16) follows: Let U (s) = max(|u(s)|, |v(s)|) , then Now we estimate the first term of the right hand side of (3.17) using the Hölder and the Sobolev inequalities and obtain that Therefore, by (3.17), (3.18), we obtain from (3.16) that By the Gagliardo-Nirenberg inequality, we obtain from (3.19) that |w(s, y)| r dy ds.
To obtain the critical existence of the weak solutions, the L ∞ a priori estimate for the weak solutions is essential. For related results, see ( [1,24]).
Let ζ j be a piecewise smooth function in Q j satisfying Multiplying (1.1) by |u(t, y)| β−2 u(t, y)ζ k (t, y) and integrating it in , we obtain that sup 0<s<t �w(s)� L r,ν ρ ≤ C�w(0)� L r,ν ρ , For the highest-order term, using the Hölder and the Sobolev inequalities, we obtain that Since and using (3.28), integrating (3.27) over t ∈ I j , we obtain that Let γ 3 > 0 be taken as Then, under the assumption (3.23), we estimate the first term of the right hand side of (3.29) and it cancels by the second term of the right hand side. Thus from (3.29) and using the estimate for the derivatives ζ j in (3.26), that |u(t, y)| 2p+β−2 ζ j (t, y) k dy |u(t, y)| β ζ j (t, y) k−2 |∇ζ j (t, y)| 2 dy (3.28) ∇ |u(t, y)|ζ j (t, y) k β β 2 2 dy .
We then claim that {u k (t)} k satisfies the assumption (3.1). Indeed, since u k,0 → u 0 in L r,ν ρ (�) as k → ∞ , we regard, by taking k 0 sufficiently large if necessary, that