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On the solution of the heat equation with nonlinear unbounded memory
Applications of Mathematics, Tome 30 (1985) no. 6, pp. 461-474 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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).
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).
DOI : 10.21136/AM.1985.104175
Classification : 35A05, 35K20, 35K55, 35K60
Keywords: heat equation; nonlinear unbounded memory; uniqueness; existence; boundary value problem 
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Doktor, Alexandr. On the solution of the heat equation with nonlinear unbounded memory. Applications of Mathematics, Tome 30 (1985) no. 6, pp. 461-474. doi: 10.21136/AM.1985.104175

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