Behaviour in time of solutions to fourth-order parabolic systems with time dependent coefficients

TL;DR

This paper investigates the behavior over time of solutions to nonlinear fourth-order parabolic systems with time-dependent coefficients, establishing conditions for boundedness and finite-time blow-up in bounded domains.

Contribution

It introduces conditions on source terms and domain shape that determine whether solutions remain bounded or blow up in finite time.

Findings

Derived a lower bound for the existence time of solutions.

Established conditions leading to finite-time blow-up.

Provided bounds for the blow-up time $t^*$.

Abstract

This paper deals with a class of initial-boundary value problems for nonlinear fourth order parabolic systems with time dependent coefficients in a bounded domain $\Omega \subset \mathbb{R}^N, N\geq 2$. Introducing suitable conditions on the source terms, we obtain a time interval $[0,T],$ where the solution remains bounded by deriving a lower bound $T$ of $t^*$. Moreover, we establish conditions on the shape of the spatial domain and on data sufficient to guarantee that the solution blows up in finite time $t^*$, deriving an upper bound for $t^*$.