on a second critical value for the local existence of solutions in lebesgue spaces

TL;DR

This paper establishes a new second critical value for the local existence of solutions to a class of parabolic equations in Lebesgue spaces, depending on initial data and nonlinearities.

Contribution

It introduces a novel second critical value for solution existence in Lebesgue spaces for certain parabolic equations with power nonlinearities.

Findings

Identifies a second critical value  for solution existence.

Provides conditions on initial data and nonlinearities for local solutions.

Demonstrates the critical value's role in solution existence based on initial data behavior.

Abstract

We provide new conditions for the local existence of solutions to the time-weighted parabolic equation $ u_t - \Delta u = h(t)f(u) \mbox{ in } \Omega \times (0,T),$ where $ \Omega $ is a arbitrary smooth domain, $f\in C(\mathbb{R})$, $h\in C([0,\infty))$ and $u(0)\in L^r(\Omega)$. As consequence of our results, considering a suitable behavior of the non-negative initial data, we obtain a second critical value $\rho^\star = 2r/(p-1),$ when $f(u)=u^p$ and $p> 1 + 2r/N$, which determines the existence (or not) of a local solution $u \in L^\infty((0,T), L^r(\Omega)).$