On Non-existence of Global Weak-predictable-random-field Solutions to a Class of SHEs

TL;DR

This paper demonstrates that solutions to certain stochastic heat equations with super-linear growth nonlinearities do not exist globally, highlighting conditions under which solutions cease to exist in finite time, especially under Poisson noise.

Contribution

It establishes non-existence results for solutions to a class of stochastic heat equations with non-Lipschitz nonlinearities, extending understanding beyond the Lipschitz case.

Findings

Solutions grow at most exponentially over time.

Solutions cease to exist at finite time under certain nonlinear conditions.

Non-existence results apply to both compensated and non-compensated Poisson noise cases.

Abstract

The multiplicative non-linearity term is usually assumed to be globally Lipschitz in most results on SPDEs. This work proves that the solutions fail to exist if the non-linearity term grows faster than linear growth. The global non-existence of the solution occurs for some non-linear conditions on $\sigma$ . Some precise conditions for existence and uniqueness of the solutions were stated and we have established that the solutions grow in time at most a precise exponential rate at some time interval; and if the solutions satisfy some non-linear conditions then they cease to exist at some finite time t . Our result also compares the non-existence of global solutions for both the compensated and non-compensated Poisson noise equations