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Published in:
2018 | OriginalPaper | Chapter
A Criterion for Blow Up in Finite Time of a System of 1-Dimensional Reaction-Diffusion Equations
Authors : Eugenio Guerrero, José Alfredo López-Mimbela
Published in: XII Symposium of Probability and Stochastic Processes
Publisher: Springer International Publishing
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Abstract
We give a criterion for blow up in finite time of the system of semilinear partial differential equations \(\frac {\partial u_{i}(t,x)}{\partial t}=\frac {1}{2}\frac {\partial ^{2}u_{i}\left (t,x\right )}{\partial x^{2}}+\frac {\varphi ^{\prime }_{i}\left (x\right )} {\varphi _{i}\left (x\right )}\frac {\partial u_{i}\left (t,x\right )}{\partial x}+u_{j}^{1+\beta _{i}}\left (t,x\right )\), t > 0, \(x\in \mathbb {R}\), with initial values of the form \(u_{i}\left (0,x\right )={h_{i}\left (x\right )}/{\varphi _{i}\left (x\right )}\), where \(0<\varphi _{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )\cap C^{2}\left (\mathbb {R}\right )\), \(0\leq h_{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )\), β
i
> 0 and i = 1, 2, j = 3 − i. Moreover, we find an upper bound T
∗ for the blowup time of such system which depends both on the initial values f
1, f
2, and the measures \(\mu _i(\mathrm {d} x)=\varphi _i^2(x)\,\mathrm {d} x\), i = 1, 2.