A semilinear heat equation with a localized nonlinear source and non-continuous initial data

Abstract

Communicated by W. SprößigThis paper is devoted to the study of the Cauchy problem for a semilinear heat equation with nonlinear term presenting a nonlinear source centered in a closed region of the spatial domain . We assume that R n is either a smooth bounded domain or the whole space R n , n 2. The initial data u 0 is assumed to belong to the Lebesgue space L r . /.

A diffusive logistic equation with concentrated and nonlocal sources

“…Indeed, controlling expression in a such way that it makes sense in L p (Ω) requires more efforts because of the nonlocal and concentrated nonlinear terms. Hence, our existence result cannot be yielded from that one in .…”

“…By B ( a , b ), we mean the special beta function evaluated in the pair ( a , b ), defined by Let and the open ball of radius , centered at the origin. The space , defined in [, page 1912], is the set of all functions φ ∈ L r (Ω) such that φ | ∂ B ∈ L q ( ∂ B ). It is a Banach space endowed with the norm In , the authors observed that has a trace notion on ∂ B so that W 1, r (Ω) is continuously identified with a subset of and can be isometrically identified with L r (Ω)× L q ( ∂ B ).…”

“…The space , defined in [, page 1912], is the set of all functions φ ∈ L r (Ω) such that φ | ∂ B ∈ L q ( ∂ B ). It is a Banach space endowed with the norm In , the authors observed that has a trace notion on ∂ B so that W 1, r (Ω) is continuously identified with a subset of and can be isometrically identified with L r (Ω)× L q ( ∂ B ).…”

“…Then, by a local mild solution of –, we mean a continuous function u :(0, T ]→ L r (Ω) such that holds, for some, r , T >0, and u ( x ,0)= u 0 ( x ). Therefore, the concentrated term requires estimates on boundary ∂ B in order to obtain bounds for , and this is what Ferreira and Villamizar‐Roa carefully did. More precisely, they obtain pointwise bounds for the Green's function in Ω and on ∂ B , and L r (Ω) and L q ( ∂ B )‐estimates, , for and (see Lemma hereafter).…”

On the Heat Equation with Nonlinearity and Singular Anisotropic Potential on the Boundary

“…L r 1 (Ω) × L r 2 (∂Ω)) was employed in [36] and [20] for studying weak solutions for an elliptic and parabolic PDE in bounded domains Ω, respectively. Let us observe that |x ′ | −1 L r 2 (∂R n + ) = ∞ for all 1 ≤ r 2 ≤ ∞ (compare with (1.10)) which prevents the use of the spaces of [20,36] for our purposes.…”