Existence to nonlinear parabolic problems with unbounded weights

Abstract

We consider the weighted parabolic problem of the type \begin{equation*} \begin{split} \left\{\begin{array}{ll} u_t-\mathrm{div}(ω_2(x)|\nabla u|^{p-2} \nabla u )= λω_1(x) |u|^{p-2}u,& x\inΩ, u(x,0)=f(x),& x\inΩ, u(x,t)=0,& x\in\partialΩ,\ t>0, \end{array}\right. \end{split} \end{equation*} for quite a general class of possibly unbounded weights ${\omega }_1,{\omega }_2$ satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that $\lambda$ is smaller than the optimal constant in the inequality.

References in corpus (1)

• Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods

Cited by in corpus (2)

• A pocket guide to nonlinear differential equations in the Musielak--Orlicz spaces
• Existence of Weak Solutions for $p(.)$-Laplacian Equation via Compact Embeddings of the Double Weighted Variable Exponent Sobolev Spaces