The well-posedness problem of anisotropic parabolic equation with variable exponents is studied in this paper. The weak solutions and the strong solutions are introduced, respectively. By a generalized Gronwall inequality, the stability of strong solutions to this equation is established, and the uniqueness of weak solutions is proved. Compared with the related works, a new boundary value condition, , is introduced the first time and has been proved that it can take place of the Dirichlet boundary value condition in some way.

中文翻译：

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各向同性抛物线方程的边值条件

研究了具有指数变化的各向异性抛物方程的适定性问题。分别介绍了弱解和强解。通过广义的Gronwall不等式，建立了该方程强解的稳定性，并证明了弱解的唯一性。与相关作品相比，一个新的边值条件，，介绍了第一次已经被证明，它可以发生在Dirichlet边界值条件的以某种方式。