# On the solution of the heat equation with nonlinear unbounded memory

Aplikace matematiky (1985)

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

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topDoktor, 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

top- A. Doktor, Heat transmission and mass transfer in hardening concrete, (In Czech), Research report III-2-3/04-05, VÚM, Praha 1983. (1983)
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- K. Rektorys, The method of discretization in time and partial differential equations, Reidel Co, Dodrecht, Holland 1982. (1982) Zbl0522.65059MR0689712
- A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, IMC. 1964. (1964) Zbl0144.34903MR0181836
- O. A. Ladyženskaja. V. A. Solonnikov N. N. Uralceva, Linear and nonlinear equations of parabolic type, (In Russian). Moskva 1967. (1967)
- 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
- G. Duvaut J. L. Lions, Inequalities in mechanics and physics, Springer, Berlin 1976. (1976) Zbl0331.35002MR0521262
- 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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