Global Existence of Solution for the Fisher Equation via Faedo–Galerkin’s Method

In this study, we consider the Fisher equation in bounded domains. By Faedo–Galerkin’s method and with a homogeneous Dirichlet conditions, the existence of a global solution is proved.

The Galerkin Method for Fourth-Order Equation of the Moore–Gibson–Thompson Type with Integral Condition

In this manuscript, we consider the fourth order of the Moore–Gibson–Thompson equation by using Galerkin’s method to prove the solvability of the given nonlocal problem.

The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space {H_{0}^{1}(\Omega)}, the Banach Sobolev space {W^{1,q}_{0}(\Omega)}, {1<q<{\infty}}, is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the {W^{1,q}_{0}(\Omega)}-{W_{0}^{1,q^{\prime}}(\Omega)} functional setting, {\frac{1}{q}+\frac{1}{q^{\prime}}=1}. The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkin’s method in this setting, for a diffusion-dominated case and under the assumption of {W^{1,q^{\prime}}}-stability of the {H_{0}^{1}}-projector.