Existence of global solutions to a semilinear pseudo-parabolic equation

TL;DR

This paper proves the global existence, decay, and blow-up conditions for solutions to a semilinear pseudo-parabolic heat equation with logarithmic and polynomial nonlinearities across all dimensions.

Contribution

It establishes the existence, decay rates, and blow-up criteria for solutions to a new class of semilinear pseudo-parabolic equations with combined nonlinearities.

Findings

Global solutions exist for all dimensions and p>1.

Solutions decay exponentially over time.

Conditions for solution blow-up are identified.

Abstract

In this article, we consider a semilinear pseudo parabolic heat equation with the nonlinearity which is the product of logarithmic and polynomial functions. Here we prove the global existence of solution to the problem for arbitrary dimension $n \geq 1$ and power index $p>1$. Asymptotic behaviour of the solution has been addressed at different energy levels. Moreover, we prove that the global solution indeed decays with an exponential rate. Finally, sufficient conditions are provided under which blow up of solutions take place.