Well-posedness and regularity for seminlinear time-dependent second and fourth order in space equations

TL;DR

This paper provides a unified convergence analysis for semilinear time-dependent PDEs of second and fourth order, establishing existence and uniqueness of solutions with both smooth and rough initial data.

Contribution

It introduces a novel, unified approach to analyze convergence and regularity for second and fourth order semilinear PDEs with Dirichlet boundary conditions.

Findings

Proves existence and uniqueness of weak solutions for smooth initial data.

Extends existence results to problems with rough initial data.

Uses Faedo-Galerkin approximation and compactness estimates for analysis.

Abstract

This article discusses a unified convergence analysis of the semilinear time-dependent equation $\partial_t u + (-1)^\mathrm{m}\Delta^{\mathrm{m}}u + u^3 - u = f$ with $\mathrm{m} \in \{1,2\}$ and homogeneous Dirichlet boundary conditions. The analysis relies on Faedo-Galerkin approximation and convergence via compactness estimates. The existence and uniqueness of the weak solution is proved when the initial data is smooth. A refined and novel analysis extends the existence result to problems with rough initial data also.