Blow-Up of the Solution for some Higher Order Hyperbolic and Neutral Evolution Systems

2011 ◽  
Vol 52-54 ◽  
pp. 121-126
Author(s):  
Ning Chen ◽  
Ji Qian Chen
Keyword(s):  
Evolution Equation ◽  
Mixed Problem ◽  
Finite Time ◽  
Blow Up ◽  
Higher Order ◽  
Neutral Evolution ◽  
Parabolic Evolution Equation ◽  
Evolution Systems

In this paper, we give some results on the blow-up behaviors of the solution to the mixed problem for some higher-order nonlinear hyperbolic and parabolic evolution equation in finite time. By introducing the “ blow-up factor ’’, we get some new conclusions, which generalize some results [4]-[5] , [6] .

scholarly journals Blow-Up Phenomena for Certain Nonlocal Evolution Equations and Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Evolution Equation ◽  
Positive Solutions ◽  
Evolution Equations ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Duality Argument ◽  
Nonlocal Evolution Equation ◽  
Global Nonexistence ◽  
Nonlocal Evolution Equations ◽  
Evolution Systems

We provide sufficient conditions for the nonexistence of global positive solutions to the nonlocal evolution equationutt(x,t)=(J ∗ u-u)(x,t)+up(x,t), (x,t)∈RN×(0,∞),(u(x,0),ut(x,0))=(u0(x),u1(x)),x∈RN,whereJ:RN→R+,p>1, and(u0,u1)∈Lloc1(RN;R+)×Lloc1(RN;R+). Next, we deal with global nonexistence for certain nonlocal evolution systems. Our method of proof is based on a duality argument.

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.

scholarly journals Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yongbin Wang ◽  
Binhua Feng
Keyword(s):  
Global Existence ◽  
Finite Time ◽  
Critical Case ◽  
Blow Up ◽  
Critical Mass ◽  
Choquard Equation ◽  
Power Type ◽  
Scale Invariant ◽  
Nonlinear Schrödinger ◽  
Sharp Thresholds

AbstractIn this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation $$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}\bigl(I _{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) \vert \psi \vert ^{p_{2}-2}\psi . $$iψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ. We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the $L^{2}$L2-critical case. Our obtained results extend and improve some recent results.

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