Unilateral problems for the wave equation with degenerate and localized nonlinear damping: well-posedness and non-stability results

2018 ◽  
Vol 291 (8-9) ◽  
pp. 1216-1239
Author(s):  
A. D. D. Cavalcanti ◽  
M. M. Cavalcanti ◽  
L. H. Fatori ◽  
M. A. Jorge Silva
Keyword(s):  
Wave Equation ◽  
Nonlinear Damping ◽  
Well Posedness ◽  
Unilateral Problems ◽  
Stability Results

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>

scholarly journals On the Parabolic and Hyperbolic Liouville Equations

2021 ◽  
Author(s):  
Tadahiro Oh ◽  
Tristan Robert ◽  
Yuzhao Wang
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Gibbs Measures ◽  
Nonlinear Heat Equation ◽  
Well Posedness ◽  
Exponential Nonlinearity ◽  
Definite Structure ◽  
Beta 2 ◽  
Moment Bounds

AbstractWe study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $$\lambda \beta e^{\beta u }$$ λ β e β u , forced by an additive space-time white noise. (i) We first study SNLH for general $$\lambda \in {\mathbb {R}}$$ λ ∈ R . By establishing higher moment bounds of the relevant Gaussian multiplicative chaos and exploiting the positivity of the Gaussian multiplicative chaos, we prove local well-posedness of SNLH for the range $$0< \beta ^2 < \frac{8 \pi }{3 + 2 \sqrt{2}} \simeq 1.37 \pi $$ 0 < β 2 < 8 π 3 + 2 2 ≃ 1.37 π . Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case $$\lambda >0$$ λ > 0 , we exploit a certain sign-definite structure in the equation and the positivity of the Gaussian multiplicative chaos. This allows us to prove global well-posedness of SNLH for the range: $$0< \beta ^2 < 4\pi $$ 0 < β 2 < 4 π . (iii) As for SdNLW in the defocusing case $$\lambda > 0$$ λ > 0 , we go beyond the Da Prato-Debussche argument and introduce a decomposition of the nonlinear component, allowing us to recover a sign-definite structure for a rough part of the unknown, while the other part enjoys a stronger smoothing property. As a result, we reduce SdNLW into a system of equations (as in the paracontrolled approach for the dynamical $$\Phi ^4_3$$ Φ 3 4 -model) and prove local well-posedness of SdNLW for the range: $$0< \beta ^2 < \frac{32 - 16\sqrt{3}}{5}\pi \simeq 0.86\pi $$ 0 < β 2 < 32 - 16 3 5 π ≃ 0.86 π . This result (translated to the context of random data well-posedness for the deterministic nonlinear wave equation with an exponential nonlinearity) solves an open question posed by Sun and Tzvetkov (2020). (iv) When $$\lambda > 0$$ λ > 0 , these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on $$\beta $$ β as in (ii) and (iii) above, we prove almost sure global well-posedness (in particular for SdNLW) and invariance of the Gibbs measures in both the parabolic and hyperbolic settings. (v) In Appendix, we present an argument for proving local well-posedness of SNLH for general $$\lambda \in {\mathbb {R}}$$ λ ∈ R without using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range $$0< \beta ^2 < \frac{4}{3} \pi \simeq 1.33 \pi $$ 0 < β 2 < 4 3 π ≃ 1.33 π , slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.

scholarly journals Boundary Stabilization of a Semilinear Wave Equation with Variable Coefficients under the Time-Varying and Nonlinear Feedback

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Bei Gong ◽  
Xiaopeng Zhao
Keyword(s):  
Wave Equation ◽  
Riemannian Geometry ◽  
Nonlinear Term ◽  
Variable Coefficients ◽  
Time Varying ◽  
Nonlinear Feedback ◽  
Boundary Stabilization ◽  
Semilinear Wave Equation ◽  
The Stability ◽  
Stability Results

We study the boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback. By the Riemannian geometry methods, we obtain the stability results of the system under suitable assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term.

Error feedback regulation for 1D anti-stable wave equation

2021 ◽  
pp. 014233122110592
Author(s):  
Zhiyuan Li ◽  
Feng-Fei Jin
Keyword(s):  
Wave Equation ◽  
Tracking Error ◽  
Feedback Regulation ◽  
Original System ◽  
Well Posedness ◽  
Error Feedback ◽  
One Dimensional ◽  
Internal Loop ◽  
Finite Dimensional ◽  
Uniformly Bounded

This paper is concerned with the boundary error feedback regulation for a one-dimensional anti-stable wave equation with distributed disturbance generated by a finite-dimensional exogenous system. Transport equation and regulator equation are introduced first to deal with the anti-damping on boundary and the distributed disturbance of the original system. Then, the tracking error and its derivative are measured to design an observer for both exosystem and auxiliary partial differential equation (PDE) system to recover the state. After proving the well-posedness of the regulator equations, we propose an observer-based controller to regulate the tracking error to zero exponentially and keep the states of all the internal loop uniformly bounded. Finally, some numerical simulations are presented to validate the effectiveness of the proposed controller.

Sign in / Sign up

Export Citation Format

Share Document