An energy analysis of degenerate hyperbolic partial differential equations.
William Layton · Applications of Mathematics · 1984
An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region$\Omega$ (E) $(tu_t)_t=\sum_{i,j=1}(a_{ij}(x)u_{x_i})_{x_j} - {a_0(x)u+f(u)}$, subject to the initial and boundary conditions, $u=0$ on $\partial\Omega$ and $u(x,0)=u_0$. (E) is degenerate at $t=0$ and thus, even in the case $f\equiv 0$, time derivatives of $u$ will blow up as $t\rightarrow 0$. Also, in the case where $f$ is locally Lipschitz, solutions of (E) can blow up for $t>0$ in finite time. Stability and convergence of the scheme in $W^{2,1}$ is shown in the linear case without assuming $u_{tt}$ (which can blow up as $t\rightarrow 0$ is smooth. Convergence of the approximation to $u$ is shown in the case where $f$ is nonlinear and locally Lipschitz. The convergence occurs in regions where $u(x,t)$ exists and is smooth. Rates of convergence are given.