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Weak self-adjointness and conservation laws for a porous medium equation. (English) Zbl 1335.35117

Summary: The concepts of self-adjoint and quasi self-adjoint equations were introduced by N. H. Ibragimov [J. Math. Anal. Appl. 318, No. 2, 742–757 (2006; Zbl 1102.34002), ibid. 333, 329–346 (2007; Zbl 1117.83127)] and [J. Phys. A, Math. Theor. 44, No. 43, Article ID 432002, 8 p. (2011; Zbl 1270.35031)]. In [Ibragimov, 2007 (loc. cit)] a general theorem on conservation laws was proved. In [J. Phys. A, Math. Theor. 44, No. 26, Article ID 262001, 6 p. (2011; Zbl 1223.35203)] we generalized the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. In this paper we find the subclasses of weak self-adjoint porous medium equations. By using the property of weak self-adjointness we construct some conservation laws associated with symmetries of the differential equation.

MSC:

35K57 Reaction-diffusion equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35K59 Quasilinear parabolic equations
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