Stochastic pseudo-parabolic equation with logarithmic nonlinearity

Yang Cao; Chengyuan Qu; Benhui Wang; Yang Cao; Chengyuan Qu; Benhui Wang

doi:10.3934/cam.2026010

Communications in Analysis and Mechanics

2026, Volume 18, Issue 1: 245-272. doi: 10.3934/cam.2026010

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Research article

Stochastic pseudo-parabolic equation with logarithmic nonlinearity

1.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China
2.
School of Mathematical Sciences, Dalian Minzu University, Dalian 116600, P. R. China

Received: 04 August 2025 Revised: 22 October 2025 Accepted: 04 January 2026 Published: 23 March 2026
35K70, 35R60

This manuscript was dedicated to exploring the existence and uniqueness of global solutions pertaining to a specific class of stochastic pseudo-parabolic equations. These equations have logarithmic nonlinearity and are driven by Brownian motion. By utilizing the Galerkin method, Prokhorov's theorem, and Skorohod's embedding theorem, we proved the existence of a global solution in the weak probabilistic sense. Subsequently, by leveraging the uniqueness of solutions and applying the Yamada-Watanabe theorem, the global existence and uniqueness of a probability strong solution were established. A notable finding, in contrast to the stochastic heat equation, was that as the pseudo-parabolic coefficient $ \mu $ increased, the growth condition imposed on the noise coefficient within the stochastic pseudo-parabolic equation could be significantly relaxed.
- stochastic pseudo-parabolic equation,
- logarithmic nonlinearity,
- global solution,
- Galerkin method,
- non-local
Citation: Yang Cao, Chengyuan Qu, Benhui Wang. Stochastic pseudo-parabolic equation with logarithmic nonlinearity[J]. Communications in Analysis and Mechanics, 2026, 18(1): 245-272. doi: 10.3934/cam.2026010

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Abstract

This manuscript was dedicated to exploring the existence and uniqueness of global solutions pertaining to a specific class of stochastic pseudo-parabolic equations. These equations have logarithmic nonlinearity and are driven by Brownian motion. By utilizing the Galerkin method, Prokhorov's theorem, and Skorohod's embedding theorem, we proved the existence of a global solution in the weak probabilistic sense. Subsequently, by leveraging the uniqueness of solutions and applying the Yamada-Watanabe theorem, the global existence and uniqueness of a probability strong solution were established. A notable finding, in contrast to the stochastic heat equation, was that as the pseudo-parabolic coefficient $ \mu $ increased, the growth condition imposed on the noise coefficient within the stochastic pseudo-parabolic equation could be significantly relaxed.

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