Blow-up problems for a parabolic equation coupled with superlinear source and local linear boundary dissipation

TL;DR

This paper investigates conditions under which solutions to a specific parabolic PDE with superlinear source and boundary dissipation blow up in finite time, providing bounds and existence results for high-energy initial data.

Contribution

It establishes sufficient conditions for finite time blow-up and constructs solutions with arbitrarily high initial energy, extending understanding of blow-up phenomena in such equations.

Findings

Finite time blow-up occurs under certain conditions.

Existence of high-energy initial data leading to blow-up.

Bounds on the blow-up time are derived.

Abstract

In this paper, we consider the finite time blow-up results for a parabolic equation coupled with superlinear source term and local linear boundary dissipation. Using a concavity argument, we derive the sufficient conditions for the solutions to blow up in finite time. In particular, we obtain the existence of finite time blow-up solutions with arbitrary high initial energy. We also derive the upper bound and lower bound of the blow up time.