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On the solution of the heat equation with nonlinear unbounded memory. (English) Zbl 0602.35056

The author studies the uniqueness and existence of the solution of the boundary value problem for the following system of nonlinear partial differential equations \[ \rho (x,t)\partial u/\partial t=div(\lambda (x,t)\text{grad} u)+q(\tau)\psi (u,\tau),\quad \partial \tau /\partial t=\psi (u(x,t),\tau (x,t)). \]
Reviewer: T.A.Dzhangveladze

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35K20 Initial-boundary value problems for second-order parabolic equations
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References:

[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.
[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. · Zbl 0522.65059
[5] A. Friedman: Partial differential equations of parabolic type. Prentice-Hall, IMC. 1964. · 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. · Zbl 0222.47011
[8] G. Duvaut J. L. Lions: Inequalities in mechanics and physics. Springer, Berlin 1976. · Zbl 0331.35002
[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
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