Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy

2018 ◽  
Vol 83 ◽  
pp. 176-181 ◽  
Author(s):  
Runzhang Xu ◽  
Xingchang Wang ◽  
Yanbing Yang
Keyword(s):  
Parabolic Equations ◽  
Initial Energy ◽  
Blowup Time

scholarly journals Semilinear pseudo-parabolic equations on manifolds with conical singularities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yitian Wang ◽  
Xiaoping Liu ◽  
Yuxuan Chen
Keyword(s):  
Energy Level ◽  
Parabolic Equations ◽  
Finite Time ◽  
Invariant Manifolds ◽  
Initial Energy ◽  
Energy Estimates ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Potential Well ◽  
Blowup Time

<p style='text-indent:20px;'>This paper studies the well-posedness of the semilinear pseudo-parabolic equations on manifolds with conical degeneration. By employing the Galerkin method and performing energy estimates, we first establish the local-in-time well-posedness of the solution. Moreover, to reveal the relationship between the initial datum and the global-in-time well-posedness of the solution we divide the initial datum into three classes by the potential well depth, i.e., the sub-critical initial energy level, the critical initial energy level and the sup-critical initial energy level (included in the arbitrary high initial energy case), and finally we give an affirmative answer to the question whether the solution exists globally or not. For the sub-critical and critical initial energy, thanks to the potential well theory, we not only obtain the invariant manifolds, global existence and asymptotic behavior of solutions, but also prove the finite time blow up of solutions and estimate the lower bound the of blowup time. For the sup-critical case, we show the assumptions for initial datum which cause the finite time blowup of the solution, realized by introducing a new auxiliary function. Additionally, we also provide some results concerning the estimates of the upper bound of the blowup time in the sup-critical initial energy.</p>

scholarly journals Global existence and nonexistence for a class of finitely degenerate coupled parabolic systems with high initial energy level

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yuxuan Chen ◽  
Jiangbo Han
Keyword(s):  
Energy Level ◽  
Global Existence ◽  
Blow Up ◽  
Parabolic Systems ◽  
Initial Energy ◽  
Blowup Time ◽  
Positive Initial Energy ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions ◽  
Coupled Parabolic Systems

<p style='text-indent:20px;'>In this paper, we consider a class of finitely degenerate coupled parabolic systems. At high initial energy level <inline-formula><tex-math id="M1">\begin{document}$ J(u_{0})&gt;d $\end{document}</tex-math></inline-formula>, we present a new sufficient condition to describe the global existence and nonexistence of solutions for problem (1)-(4) respectively. Moreover, by applying the Levine's concavity method, we give some affirmative answers to finite time blow up of solutions at arbitrary positive initial energy <inline-formula><tex-math id="M2">\begin{document}$ J(u_{0})&gt;0 $\end{document}</tex-math></inline-formula>, including the estimate of upper bound of blowup time.</p>

scholarly journals Global nonexistence and blow-up results for a quasi-linear evolution equation with variable-exponent nonlinearities

2021 ◽  
Vol 66 (3) ◽  
pp. 553-566
Author(s):  
Abita Rahmoune ◽  
Benyattou Benabderrahmane
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Nonnegative Function ◽  
Initial Energy ◽  
Positive Energy ◽  
Linear Evolution ◽  
Global Nonexistence ◽  
Linear Evolution Equation ◽  
Linear Parabolic Equations ◽  
The Given

"In this paper, we consider a class of quasi-linear parabolic equations with variable exponents, $$a\left( x,t\right) u_{t}-\Delta _{m\left( .\right) }u=f_{p\left( .\right)}\left( u\right)$$ in which $f_{p\left( .\right)}\left( u\right)$ the source term, $a(x,t)>0$ is a nonnegative function, and the exponents of nonlinearity $m(x)$, $p(x)$ are given measurable functions. Under suitable conditions on the given data, a finite-time blow-up result of the solution is shown if the initial datum possesses suitable positive energy, and in this case, we precise estimate for the lifespan $T^{\ast }$ of the solution. A blow-up of the solution with negative initial energy is also established."

scholarly journals Global Existence, Exponential Decay and Blow-Up of Solutions for a Class of Fractional Pseudo-Parabolic Equations with Logarithmic Nonlinearity

2018 ◽  
Author(s):  
Wenjun Liu ◽  
Jiangyong Yu ◽  
Gang Li
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Global Weak Solution ◽  
Energy Functional ◽  
Initial Energy ◽  
Invariant Sets ◽  
Potential Wells ◽  
Fractional Sobolev Space ◽  
The Relationship ◽  
Fractional Laplace Operator

In this paper, we study the fractional pseudo-parabolic equations ut + (&minus;△)s u + (&minus;△)s ut = u log |u|. Firstly, we recall the relationship between the fractional Laplace operator (&minus;△)s and the fractional Sobolev space Hs and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of global weak solution: for the low initial energy J(u0) &lt; d, the solution is global in time with I(u0) &gt; 0 or ∥u0∥X0(&Omega;) = 0 and blows up +&infin; with I(u0) &lt; 0; for the critical initial energy J(u0) = d, the solution is global in time with I(u0) &ge; 0 and blows up at +&infin; with I(u0) &lt; 0. The decay estimate of the energy functional for the global solution is also given.

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