Weak self-adjoint differential equations. (English) Zbl 1223.35203

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)]. In [N. H. Ibragimov, ibid. 333, 329–346 (2007; Zbl 1117.83127)], a general theorem on conservation laws was proved. In this paper, we generalize the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. We find a class of weak self-adjoint quasilinear parabolic equations. The property of a differential equation to be weak self-adjoint is important for constructing conservation laws associated with symmetries of the differential equation.


35K59 Quasilinear parabolic equations
35B06 Symmetries, invariants, etc. in context of PDEs
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