Finite element analysis of a nonlinear heat Equation with damping and pumping effects

Abstract

We study the following nonlinear heat equation with damping and pumping effects (a reaction-diffusion equation) posed on a bounded simply connected convex domain $\Omega \subset {\mathbb{R}}^d$, $d\ge 1$ with Lipschitz boundary $\partial \Omega$: $$\frac{\partial u(t)}{\partial t}-\nu \Delta u(t)+\alpha |u(t){|}^{p-2}u(t)-\sum _{\ell =1}^M{\beta }_{\ell }|u(t){|}^{q_{\ell }-2}u(t)=f(t),t>0,$$ subject to homogeneous Dirichlet boundary conditions and the initial condition $u(0)=u_0$, where $2\le p<\infty$ and $2\le q_{\ell }<p$ for $1\le \ell \le M$. For $u_0\in L^2(\Omega )$ and $f\in L^2(0,T;H^{-1}(\Omega ))$, we establish the existence and uniqueness of a weak solution for all dimensions $d\in \mathbb{N}$ and damping exponents $2\le p<\infty$. Furthermore, for $u_0\in H^2(\Omega )\cap H_0^1(\Omega )$ and $f\in H^1(0,T;H^1(\Omega ))$, we obtain regularity results: these hold for every $2\le p<\infty$ when $1\le d\le 4$, and for $2\le p\le \frac{2d-6}{d-4}$ when $d\ge 5$. We further conduct finite element analysis using conforming, nonconforming, and discontinuous Galerkin methods, deriving a priori error estimates for both semi- and fully discrete schemes, supported by numerical results. To relax restrictions on $p$ in the semidiscrete analysis, we use appropriate projection/interpolation operators: the Ritz projection in the conforming case ($2\le p\le \frac{2d}{d-2}$), the Scott-Zhang interpolation for $\frac{2d}{d-2}<p\le \frac{2d-6}{d-4}$, the Clément interpolation in the nonconforming setting, and the $L^2$-projection in the DG framework. In the fully discrete case, error estimates hold for the above $p$-range under $u_0\in D(A^{3/2})$ and $f\in H^1(0,T;H^1(\Omega ))$.