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heat equation; nonlinear unbounded memory; uniqueness; existence; boundary value problem
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).
[1] A. Doktor: Heat transmission and mass transfer in hardening concrete. (In Czech), Research report III-2-3/04-05, VÚM, Praha 1983.
[2] E. Rastrup: Heat of hydration of conrete. Magazine of Concrete Research, v. 6, no 17, 1954. DOI 10.1680/macr.1954.6.17.79
[3] K. Rektorys : Nonlinear problem of heat conduction in concrete massives. (In Czech), Thesis MÚ ČSAV, Praha 1961.
[4] K. Rektorys: The method of discretization in time and partial differential equations. Reidel Co, Dodrecht, Holland 1982. MR 0689712 | Zbl 0522.65059
[5] A. Friedman: Partial differential equations of parabolic type. Prentice-Hall, IMC. 1964. MR 0181836 | Zbl 0144.34903
[6] O. A. Ladyženskaja. V. A. Solonnikov N. N. Uralceva: Linear and nonlinear equations of parabolic type. (In Russian). Moskva 1967.
[7] T. Kato: Linear evolution equations of "hyperbolic" type. J. Fac. Sci. Univ. Tokyo, Sec. 1, vol. XVII (1970), pyrt 182, 241-258. MR 0279626 | Zbl 0222.47011
[8] G. Duvaut J. L. Lions: Inequalities in mechanics and physics. Springer, Berlin 1976. MR 0521262
[9] A. Doktor: Mixed problem for semilinear hyperbolic equation of second order with Dirichlet boundary condition. Czech. Math. J., 23 (98), 1973, 95-122. MR 0348276 | Zbl 0255.35061
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