Existence of solution and qualitative behavior for a class of heat equations

Abstract

The objective of this paper is to investigate the existence of solutions and their qualitative behavior for a given class of nonlinear evolutionary equations. We demonstrate that the pseudo-differential operator Δexp⁡(−cΔ) acts as the infinitesimal generator of the solution operator. Here, Δ denotes the Euclidean Laplace operator, and c is a positive constant. We establish the appropriate domain for the operator Δexp⁡(−cΔ) and prove that it generates a C0 semigroup on L2(RN). Additionally, we introduce a scale of spaces wherein smooth solutions exist, and these spaces are continuously embedded into the Sobolev class. Finally, we investigate the nonlinear evolution problem for a broad class of nonlinearities.