Existence for singular doubly nonlinear systems of porous medium type with time dependent boundary values

Abstract

In this paper we prove the existence of variational solutions to the Cauchy–Dirichlet problem with time dependent boundary values associated with doubly nonlinear systems

$$\begin{aligned} \partial _t \big (|u|^{m-1}u\big ) - {{\,\mathrm{div}\,}}(D_\xi f(Du)) = 0 \end{aligned}$$

with \(m>1\) and a convex function f satisfying a standard p-growth condition for an exponent \(p \in (1,\infty )\). The proof relies on a nonlinear version of the method of minimizing movements.

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The author has been supported by the Studienstiftung des deutschen Volkes

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Correspondence to Leah Schätzler.

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Schätzler, L. Existence for singular doubly nonlinear systems of porous medium type with time dependent boundary values. J Elliptic Parabol Equ 5, 383–421 (2019). https://doi.org/10.1007/s41808-019-00048-7

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Keywords

  • Porous medium equation
  • Doubly nonlinear systems
  • Existence
  • Minimizing movements

Mathematics Subject Classification

  • 35K86
  • 49J40
  • 49J45