scholarly journals Semilinear viscous Moore-Gibson-Thompson equation with the derivative-type nonlinearity: Global existence versus blow-up

2021 ◽  
Vol 7 (1) ◽  
pp. 247-257
Author(s):  
Jincheng Shi ◽  
◽  
Yan Zhang ◽  
Zihan Cai ◽  
Yan Liu ◽  
...  
Keyword(s):  
Global Existence ◽  
Nonlinear Problem ◽  
Weak Solutions ◽  
Blow Up ◽  
Small Data ◽  
Estimates Of Solutions ◽  
Derivative Type

<abstract><p>In this paper, we study global existence and blow-up of solutions to the viscous Moore-Gibson-Thompson (MGT) equation with the nonlinearity of derivative-type $ |u_t|^p $. We demonstrate global existence of small data solutions if $ p &gt; 1+4/n $ ($ n\leq 6 $) or $ p\geq 2-2/n $ ($ n\geq 7 $), and blow-up of nontrivial weak solutions if $ 1 &lt; p\leq 1+1/n $. Deeply, we provide estimates of solutions to the nonlinear problem. These results complete the recent works for semilinear MGT equations by <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>.</p></abstract>

Global Existence and Blow-Up of Weak Solutions for Nonlinear Wave Equations

2014 ◽  
Vol 635-637 ◽  
pp. 1565-1568
Author(s):  
Yun Zhu Gao ◽  
Wei Guo ◽  
Tian Luan
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Weak Solutions ◽  
Wave Equations ◽  
Blow Up ◽  
Nonlinear Damping ◽  
Nonlinear Wave Equations ◽  
Potential Well ◽  
Source Terms ◽  
Damping And Source Terms

In this paper, we discuss the nonlinear wave equations with nonlinear damping and source terms. By using the potential well methods, we get a result for the global existence and blow-up of the solutions.

scholarly journals A Note on the Generalized Camassa-Holm Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Yun Wu ◽  
Ping Zhao
Keyword(s):  
Boundary Condition ◽  
Global Existence ◽  
Weak Solutions ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Blow Up ◽  
Holm Equation ◽  
Special Cases ◽  
Novikov Equation

We study the generalized Camassa-Holm equation which contains the Camassa-Holm (CH) equation and Novikov equation as special cases with the periodic boundary condition. We get a blow-up scenario and obtain the global existence of strong and weak solutions under suitable assumptions, respectively. Then, we construct the periodic peaked solutions and apply them to prove the ill-posedness inHswiths<3/2.

scholarly journals Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations

2020 ◽  
Vol 9 (1) ◽  
pp. 1569-1591
Author(s):  
Menglan Liao ◽  
Qiang Liu ◽  
Hailong Ye
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Finite Time ◽  
Evolution Equations ◽  
Blow Up ◽  
Initial Energy ◽  
Potential Well

Abstract In this paper, we study the fractional p-Laplacian evolution equation with arbitrary initial energy, $$\begin{array}{} \displaystyle u_t(x,t) + (-{\it\Delta})_p^s u(x,t) = f(u(x,t)), \quad x\in {\it\Omega}, \,t \gt 0, \end{array} $$ where $\begin{array}{} (-{\it\Delta})_p^s \end{array} $ is the fractional p-Laplacian with $\begin{array}{} p \gt \max\{\frac{2N}{N+2s},1\} \end{array} $ and s ∈ (0, 1). Specifically, by the modified potential well method, we obtain the global existence, uniqueness, and blow-up in finite time of the weak solution for the low, critical and high initial energy cases respectively.

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