| Title:
|
An energy analysis of degenerate hyperbolic partial differential equations. (English) |
| Author:
|
Layton, William J. |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
29 |
| Issue:
|
5 |
| Year:
|
1984 |
| Pages:
|
350-366 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| Summary lang:
|
Russian |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
degenerate equation |
| Keyword:
|
Lipschitz |
| Keyword:
|
energy analysis |
| Keyword:
|
semi-discrete Galerkin method |
| Keyword:
|
semilinear equation |
| Keyword:
|
stability |
| Keyword:
|
convergence |
| MSC:
|
35L10 |
| MSC:
|
35L80 |
| MSC:
|
65M60 |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 0564.65073 |
| idMR:
|
MR0772270 |
| DOI:
|
10.21136/AM.1984.104105 |
| . |
| Date available:
|
2008-05-20T18:25:44Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104105 |
| . |
| Reference:
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