WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 14, 2015
The Solution of a Parabolic Equation with Unbounded Flux Term
Authors: , ,
Abstract: Consider Cauchy problem of the degenerate parabolic equation
$$\frac{\partial u}{\partial t} = \frac{\partial }{\partial x_{i}} \left ( a^{ij} (u)\frac{\partial u}{\partial x_{i}}\right ) + div(uE).$$
A new kind of entropy solution is introduced, which is stronger than the general one. Supposing that $$u_{0} ∈ L^{∞}(\textbf R^{N} ), E = {E_{i}}, E_{i} ∈ E^{2}$$,
by a modified regularization method, the problem is translated into a approximate
Cauchy problem. By choosing suitable testing functions, the BV estimates of the solutions of the approximate
Cauchy problem are obtained. According to Kolomogroff’s Theorem, a convergent subsequence can be extracted,
then the existence of the entropy solution of the original Cauchy problem is obtained. At last, by Kruzkov bivariables method, the stability of the entropy solutions is obtained, provided that $$E_{i}x_{i} ≥ 0.$$
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Pages: 116-126
WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #12