An Exterior Parabolic Differential Inequality Under Semilinear Dynamical Boundary Conditions

Abstract

We study the existence and nonexistence of global weak solutions to the semilinear parabolic differential inequality

$$\begin{aligned} \partial _t u-\Delta u \ge |u|^p,\quad (t,x)\in (0,\infty )\times B^c, \end{aligned}$$

where \(p>1\), B is the closed unit ball in \({\mathbb {R}}^N\) (\(N\ge 2\)) and \(B^c\) is its complement, under the semilinear dynamical boundary conditions

$$\begin{aligned} \partial _t u+u \ge |u|^q +w(x), \quad (t,x)\in (0,\infty )\times \partial B \end{aligned}$$

or

$$\begin{aligned} \partial _t u+\partial _\nu u +\alpha u \ge |u|^q +w(x), \quad (t,x)\in (0,\infty )\times \partial B, \end{aligned}$$

where \(q>1\), \(\alpha \ge 0\), \(\partial _\nu :=\frac{\partial }{\partial \nu ^+}\), \(\nu ^+\) is the outward unit normal (relative to \(B^c\)) on \(\partial B\) and \(w\in L^1(\partial B)\), \(\int _{\partial B} w(x)\,\hbox {d}S_x\ge 0\). The cases \(\int _{\partial B} w(x)\,\hbox {d}S_x= 0\) and \(\int _{\partial B} w(x)\,\hbox {d}S_x>0\) are discussed separately.

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Acknowledgements

M. Jleli is supported by Researchers Supporting Project Number (RSP-2019/57), King Saud University, Riyadh, Saudi Arabia.

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Correspondence to Bessem Samet.

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Communicated by Yong Zhou.

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Alqahtani, A., Jleli, M., Kirane, M. et al. An Exterior Parabolic Differential Inequality Under Semilinear Dynamical Boundary Conditions. Bull. Malays. Math. Sci. Soc. 44, 639–660 (2021). https://doi.org/10.1007/s40840-020-00973-0

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Keywords

  • Semilinear parabolic equation
  • Semilinear dynamical boundary conditions
  • Exterior domain

Mathematics Subject Classification

  • 35K58
  • 35B44
  • 35B33