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Finite Time Blow-up for a Non-linear Parabolic Equation with a Gradient Term and Applications

✍ Scribed by Philippe Souplet

Publisher
John Wiley and Sons
Year
1996
Tongue
English
Weight
732 KB
Volume
19
Category
Article
ISSN
0170-4214

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✦ Synopsis

We give new finite time blow-up results for the non-linear parabolic equations u, -Au = up and u, -Au + pIVul4 = up. We first establish an a priori bound in Lpf ' for the positive non-decreasing global solutions. As a consequence, we prove in particular that for the second equation on RN, with q = 2p/(p + 1) and small p > 0, blow-up can occur for any N 2 1, p > 1, ( N -2)p < N + 2 and without energy restriction on the initial data. Incidentally, we present a simple model in population dynamics involving this equation.

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## Abstract We consider the blowup of solutions of the initial boundary value problem for a class of non‐linear evolution equations with non‐linear damping and source terms. By using the energy compensation method, we prove that when __p__>max{__m__, __Ξ±__}, where __m__, __Ξ±__ and __p__ are non‐neg