Linear and nonlinear heat equations in $L^q_\delta$ spaces and universal bounds for global solutions

Abstract

We develop a theory of both linear and nonlinear heat equations in the weighted Lebesgue spaces \(L^q_\delta\), where \(\delta\) is the distance to the boundary. In particular, we prove an optimal \(L^q_\delta-L^r_\delta\) estimate for the heat semigroup, and we establish sharp results on local existence-uniqueness and local nonexistence of solutions for semilinear heat equations with initial values in those spaces. This theory enables us to obtain new types of results concerning positive global solutions of superlinear parabolic problems. Namely, under certain assumptions, we prove that any global solution is uniformly bounded for \(t\geq \tau>0\) by a universal constant, independent of the initial data. In all previous results, the bounds for global solutions were depending on the initial data.