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

This article discusses a unified convergence analysis of the semilinear time-dependent equation ${\mathrm{\partial }}_{t}u+(-1{)}^{\mathrm{m}}{\mathrm{\Delta }}^{\mathrm{m}}u+u^3-u=f$ 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.