On the solution of the heat equation with nonlinear unbounded memory

Alexander Doktor

Aplikace matematiky (1985)

  • Volume: 30, Issue: 6, page 461-474
  • ISSN: 0862-7940

Abstract

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The paper deals with the question of global solution u , τ to boundary value problem for the system of semilinear heat equation for u and complementary nonlinear differential equation for τ (“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 ( 𝒫 ) holds. The condition ( 𝒫 ) is verified for some particular cases (e. g.: bounded nonlinearity, homogeneous Neumann problem (even for unbounded nonlinearities), apriori estimate of the solution holds).

How to cite

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Doktor, Alexander. "On the solution of the heat equation with nonlinear unbounded memory." Aplikace matematiky 30.6 (1985): 461-474. <http://eudml.org/doc/15428>.

@article{Doktor1985,
abstract = {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 $(\mathcal \{P\})$ holds. The condition $(\mathcal \{P\})$ is verified for some particular cases (e. g.: bounded nonlinearity, homogeneous Neumann problem (even for unbounded nonlinearities), apriori estimate of the solution holds).},
author = {Doktor, Alexander},
journal = {Aplikace matematiky},
keywords = {heat equation; nonlinear unbounded memory; uniqueness; existence; boundary value problem; heat equation; nonlinear unbounded memory; uniqueness; existence; boundary value problem},
language = {eng},
number = {6},
pages = {461-474},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the solution of the heat equation with nonlinear unbounded memory},
url = {http://eudml.org/doc/15428},
volume = {30},
year = {1985},
}

TY - JOUR
AU - Doktor, Alexander
TI - On the solution of the heat equation with nonlinear unbounded memory
JO - Aplikace matematiky
PY - 1985
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 30
IS - 6
SP - 461
EP - 474
AB - 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 $(\mathcal {P})$ holds. The condition $(\mathcal {P})$ is verified for some particular cases (e. g.: bounded nonlinearity, homogeneous Neumann problem (even for unbounded nonlinearities), apriori estimate of the solution holds).
LA - eng
KW - heat equation; nonlinear unbounded memory; uniqueness; existence; boundary value problem; heat equation; nonlinear unbounded memory; uniqueness; existence; boundary value problem
UR - http://eudml.org/doc/15428
ER -

References

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  1. A. Doktor, Heat transmission and mass transfer in hardening concrete, (In Czech), Research report III-2-3/04-05, VÚM, Praha 1983. (1983) 
  2. E. Rastrup, Heat of hydration of conrete, Magazine of Concrete Research, v. 6, no 17, 1954. (1954) 
  3. K. Rektorys, Nonlinear problem of heat conduction in concrete massives, (In Czech), Thesis MÚ ČSAV, Praha 1961. (1961) 
  4. K. Rektorys, The method of discretization in time and partial differential equations, Reidel Co, Dodrecht, Holland 1982. (1982) Zbl0522.65059MR0689712
  5. A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, IMC. 1964. (1964) Zbl0144.34903MR0181836
  6. O. A. Ladyženskaja. V. A. Solonnikov N. N. Uralceva, Linear and nonlinear equations of parabolic type, (In Russian). Moskva 1967. (1967) 
  7. T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, Sec. 1, vol. XVII (1970), pyrt 182, 241-258. (1970) Zbl0222.47011MR0279626
  8. G. Duvaut J. L. Lions, Inequalities in mechanics and physics, Springer, Berlin 1976. (1976) Zbl0331.35002MR0521262
  9. A. Doktor, Mixed problem for semilinear hyperbolic equation of second order with Dirichlet boundary condition, Czech. Math. J., 23 (98), 1973, 95-122. (1973) Zbl0255.35061MR0348276

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