| Title:
|
Classical global solutions of the initial boundary value problems for a class of nonlinear parabolic equations (English) |
| Author:
|
Chen, Guowang |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
35 |
| Issue:
|
3 |
| Year:
|
1994 |
| Pages:
|
431-443 |
| . |
| Category:
|
math |
| . |
| Summary:
|
The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation $$ u_t=-A(t)u_{x^4}+B(t)u_{x^2}+g(u)_{x^2}+f(u)_{x}+h(u_{x})_{x}+G(u) $$ with the initial boundary value conditions $$ u(-\ell ,t)=u(\ell ,t)=0,\quad u_{x^2}(-\ell ,t)=u_{x^2}(\ell ,t)=0,\quad u(x,0)=\varphi (x), $$ or with the initial boundary value conditions $$ u_{x}(-\ell ,t)=u_{x}(\ell ,t)=0,\quad u_{x^3}(-\ell ,t)=u_{x^3}(\ell ,t)=0,\quad u(x,0)=\varphi (x), $$ are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions. (English) |
| Keyword:
|
nonlinear parabolic equation |
| Keyword:
|
initial boundary value problem |
| Keyword:
|
classical global solutions |
| MSC:
|
35B40 |
| MSC:
|
35K35 |
| MSC:
|
35K55 |
| MSC:
|
35K60 |
| idZBL:
|
Zbl 0808.35049 |
| idMR:
|
MR1307271 |
| . |
| Date available:
|
2009-01-08T18:12:21Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118684 |
| . |
| Reference:
|
[1] Cohen D.S., Murray J.D.: A generalized diffusion model for growth and dispersal in population.J. Math. Biol. 12 (1981), 237-249. MR 0624561 |
| Reference:
|
[2] Naimark M.A.: Linear Differential Operators.Moscow, 1954. Zbl 0227.34020, MR 0067292 |
| Reference:
|
[3] Maz'ja V.G.: Sobolev Spaces.Springer-Verlag, 1985. Zbl 0692.46023, MR 0817985 |
| Reference:
|
[4] Zhou Yulin, Fu Hongyuan: The nonlinear hyperbolic systems of higher order of generalized Sine-Gordon type (in Chinese).Acta Math. Sinica 26 (1983), 234-249. MR 0694886 |
| Reference:
|
[5] Chen Guowang: First boundary problems for nonlinear parabolic and hyperbolic coupled systems of higher order.Chinese Journal of Contemporary Mathematics 9 (1988), 98-116. MR 0997346 |
| Reference:
|
[6] Liu Baoping, Pao C.V.: Integral representation of generalized diffusion model in population problems.Journal of Integral Equations 6 (1984), 175-185. MR 0733043 |
| Reference:
|
[7] Chen Guowang: Initial value problem for a class of nonlinear parabolic system of fourth- order.Acta Mathematica Scientia 11 (1991), 393-400. MR 1174369 |
| . |
