scholarly journals Growth order and blow up points for the parabolic Burgers' equation under dynamical boundary conditions

2012 ◽  
Vol 6 (3) ◽  
pp. 825-836
Author(s):  
Jean-François Rault ◽  
Gaëlle Pincet Mailly ◽  
Joachim von Below
Keyword(s):  
Boundary Conditions ◽  
Burgers Equation ◽  
Blow Up ◽  
Growth Order ◽  
Dynamical Boundary Conditions

scholarly journals Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Enzo Vitillaro
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Boundary Conditions ◽  
Blow Up ◽  
Nonlinear Damping ◽  
Beltrami Operator ◽  
Well Posedness ◽  
Nonlinear Source ◽  
Global Nonexistence ◽  
Dynamical Boundary Conditions

<p style='text-indent:20px;'>The aim of this paper is to give global nonexistence and blow–up results for the problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} u_{tt}-\Delta u+P(x,u_t) = f(x,u) \qquad &amp;\text{in $(0,\infty)\times\Omega$,}\\ u = 0 &amp;\text{on $(0,\infty)\times \Gamma_0$,}\\ u_{tt}+\partial_\nu u-\Delta_\Gamma u+Q(x,u_t) = g(x,u)\qquad &amp;\text{on $(0,\infty)\times \Gamma_1$,}\\ u(0,x) = u_0(x),\quad u_t(0,x) = u_1(x) &amp; \text{in $\overline{\Omega}$,} \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded open <inline-formula><tex-math id="M2">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> subset of <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N\ge 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \Gamma = \partial\Omega $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ (\Gamma_0,\Gamma_1) $\end{document}</tex-math></inline-formula> is a partition of <inline-formula><tex-math id="M7">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \Gamma_1\not = \emptyset $\end{document}</tex-math></inline-formula> being relatively open in <inline-formula><tex-math id="M9">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \Delta_\Gamma $\end{document}</tex-math></inline-formula> denotes the Laplace–Beltrami operator on <inline-formula><tex-math id="M11">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ \nu $\end{document}</tex-math></inline-formula> is the outward normal to <inline-formula><tex-math id="M13">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>, and the terms <inline-formula><tex-math id="M14">\begin{document}$ P $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ Q $\end{document}</tex-math></inline-formula> represent nonlinear damping terms, while <inline-formula><tex-math id="M16">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ g $\end{document}</tex-math></inline-formula> are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well–posedness.</p>

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