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Blow‐up for a class of semilinear integro‐differential equations of parabolic type
Author(s) -
Hirata Daisuke
Publication year - 1999
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19990910)22:13<1087::aid-mma73>3.0.co;2-y
Subject(s) - mathematics , class (philosophy) , dirichlet boundary condition , homogeneous , type (biology) , mathematical analysis , differential equation , integro differential equation , boundary value problem , parabolic partial differential equation , first order partial differential equation , combinatorics , ecology , artificial intelligence , computer science , biology
In this paper, we study the following semilinear integro‐differential equation of the parabolic type that arise in the theory of nuclear reactor kinetics:$$u_{t}-\Delta u=\left(\int^t_0 u^p(s) {\rm d}s \right) u^q \qquad{\rm in\ } (0, T)\times\Omega,$$under homogeneous Dirichlet boundary condition, where p, q ⩾1. We first establish the local solvability of a large class of semilinear non‐local equations including the above equation. Next, we give the finite time blow‐up result by some modification of Kaplan's method and also the existence of global solutions by the comparison method. Copyright © 1999 John Wiley & Sons, Ltd.