| Title:
|
On the solution of the heat equation with nonlinear unbounded memory (English) |
| Author:
|
Doktor, Alexandr |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
30 |
| Issue:
|
6 |
| Year:
|
1985 |
| Pages:
|
461-474 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| Summary lang:
|
Russian |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper deals with the question of global solution $u,\tau$ to boundary value problem for the system of semilinear heat equation for $u$ and complementary nonlinear differential equation for $\tau$ ("thermal memory"). Uniqueness of the solution is shown and the method of successive approximations is used for the proof of existence of a global solution provided the condition $(\Cal P)$ holds. The condition $(\Cal P)$ is verified for some particular cases (e. g.: bounded nonlinearity, homogeneous Neumann problem (even for unbounded nonlinearities), apriori estimate of the solution holds). (English) |
| Keyword:
|
heat equation |
| Keyword:
|
nonlinear unbounded memory |
| Keyword:
|
uniqueness |
| Keyword:
|
existence |
| Keyword:
|
boundary value problem |
| MSC:
|
35A05 |
| MSC:
|
35K20 |
| MSC:
|
35K55 |
| MSC:
|
35K60 |
| idZBL:
|
Zbl 0602.35056 |
| idMR:
|
MR0813534 |
| DOI:
|
10.21136/AM.1985.104175 |
| . |
| Date available:
|
2008-05-20T18:28:56Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104175 |
| . |
| Reference:
|
[1] A. Doktor: Heat transmission and mass transfer in hardening concrete.(In Czech), Research report III-2-3/04-05, VÚM, Praha 1983. |
| Reference:
|
[2] E. Rastrup: Heat of hydration of conrete.Magazine of Concrete Research, v. 6, no 17, 1954. 10.1680/macr.1954.6.17.79 |
| Reference:
|
[3] K. Rektorys : Nonlinear problem of heat conduction in concrete massives.(In Czech), Thesis MÚ ČSAV, Praha 1961. |
| Reference:
|
[4] K. Rektorys: The method of discretization in time and partial differential equations.Reidel Co, Dodrecht, Holland 1982. Zbl 0522.65059, MR 0689712 |
| Reference:
|
[5] A. Friedman: Partial differential equations of parabolic type.Prentice-Hall, IMC. 1964. Zbl 0144.34903, MR 0181836 |
| Reference:
|
[6] O. A. Ladyženskaja. V. A. Solonnikov N. N. Uralceva: Linear and nonlinear equations of parabolic type.(In Russian). Moskva 1967. |
| Reference:
|
[7] T. Kato: Linear evolution equations of "hyperbolic" type.J. Fac. Sci. Univ. Tokyo, Sec. 1, vol. XVII (1970), pyrt 182, 241-258. Zbl 0222.47011, MR 0279626 |
| Reference:
|
[8] G. Duvaut J. L. Lions: Inequalities in mechanics and physics.Springer, Berlin 1976. MR 0521262 |
| Reference:
|
[9] A. Doktor: Mixed problem for semilinear hyperbolic equation of second order with Dirichlet boundary condition.Czech. Math. J., 23 (98), 1973, 95-122. Zbl 0255.35061, MR 0348276 |
| . |
