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Layton, William J.
An energy analysis of degenerate hyperbolic partial differential equations. (English). Aplikace matematiky, vol. 29 (1984), issue 5, pp. 350-366
MSC: 35L10, 35L80, 65M60, 65N30 | MR 0772270 | Zbl 0564.65073 | DOI: 10.21136/AM.1984.104105
Keywords:
degenerate equation; Lipschitz; energy analysis; semi-discrete Galerkin method; semilinear equation; stability; convergence
Summary:
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.
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