Opuscula Math. 39, no. 3 (2019), 395-414
https://doi.org/10.7494/OpMath.2019.39.3.395

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.
• Accepted: 2018-08-14.
• Published online: 2019-02-23.