Interpolating the Stochastic Heat and Wave Equations with Time-independent Noise: Solvability and Exact Asymptotics

TL;DR

This paper investigates stochastic heat and wave equations with time-independent noise, establishing conditions for global solutions, deriving exact moment asymptotics, and identifying blow-up times, thereby unifying known results for these equations.

Contribution

It provides new sharp conditions for solution existence, exact asymptotics, and blow-up times, extending and interpolating results for stochastic heat and wave equations.

Findings

Established conditions for global $L^p$-solutions.

Derived exact moment asymptotics for solutions.

Identified precise blow-up times for local solutions.

Abstract

In this article, we study a class of stochastic partial differential equations with fractional differential operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which a unique global $L^p(\Omega)$-solution exits for all $p\ge 2$. In this case, we derive exact moment asymptotics following the same strategy in a recent work by Balan et al [1]. In the case when there exits only a local solution, we determine the precise deterministic time, $T_2$, before which a unique $L^2(\Omega)$-solution exits, but after which the series corresponding to the $L^2(\Omega)$ moment of the solution blows up. By properly choosing the parameters, results in this paper interpolate the known results for both stochastic heat and wave equations.