Opuscula Math. 39, no. 3 (2019), 395-414

Opuscula Mathematica

Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term

Mitsuhiro Nakao

Abstract. We give an existence theorem of global solution to the initial-boundary value problem for \(u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)\) under some smallness conditions on the initial data, where \(\sigma (v^2)\) is a positive function of \(v^2\ne 0\) admitting the degeneracy property \(\sigma(0)=0\). We are interested in the case where \(\sigma(v^2)\) has no exponent \(m \geq 0\) such that \(\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0\). A typical example is \(\sigma(v^2)=\operatorname{log}(1+v^2)\). \(f(u)\) is a function like \(f=|u|^{\alpha} u\). A decay estimate for \(\|\nabla u(t)\|_{\infty}\) is also given.

Keywords: degenerate quasilinear parabolic equation, nonlinear source term, Moser's method.

Mathematics Subject Classification: 35B40, 35D35, 58J35, 58K30.

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  • Mitsuhiro Nakao
  • Kyushu University, Faculty of Mathematics, Moto-oka 744, Fukuoka 819-0395, Japan
  • Communicated by Marius Ghergu.
  • Received: 2018-05-28.
  • Accepted: 2018-08-14.
  • Published online: 2019-02-23.
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Cite this article as:
Mitsuhiro Nakao, Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term, Opuscula Math. 39, no. 3 (2019), 395-414, https://doi.org/10.7494/OpMath.2019.39.3.395

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