The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov (2006 J. Math. Anal. Appl. 318 742-57; 2007 Arch. ALGA 4 55-60). In Ibragimov (2007 J. Math. Anal. Appl. 333 311-28), 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 quasi-linear 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.
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机译：Ibragimov（2006 J. Math。Anal。Appl。318 742-57; 2007 Arch。ALGA 4 55-60）引入了自伴方程和拟自伴方程的概念。在Ibragimov（2007 J. Math。Anal。Appl。333 311-28）中，证明了关于守恒定律的一般性定理。在本文中，我们通过引入弱自伴方程的定义，推广了自伴方程和准自伴方程的概念。我们发现了一类弱的自伴拟线性抛物方程。微分方程具有弱的自伴性的性质对于构造与微分方程的对称性有关的守恒定律很重要。
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