scholarly journals Blow-Up of Solutions to Fractional-in-Space Burgers-Type Equations

2021 ◽  
Vol 5 (4) ◽  
pp. 249
Author(s):  
Munirah Alotaibi ◽  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Finite Time ◽  
Function Method ◽  
Burgers Equation ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Bounded Domains ◽  
Initial Value ◽  
Test Function ◽  
Burgers Type Equations ◽  
Test Function Method

We consider fractional-in-space analogues of Burgers equation and Korteweg-de Vries-Burgers equation on bounded domains. Namely, we establish sufficient conditions for finite-time blow-up of solutions to the mentioned equations. The obtained conditions depend on the initial value and the boundary conditions. Some examples are provided to illustrate our obtained results. In the proofs of our main results, we make use of the test function method and some integral inequalities.

scholarly journals Nonexistence Results for Some Classes of Nonlinear Fractional Differential Inequalities

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Function Method ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Differential Inequalities ◽  
Test Function ◽  
Fractional Differential ◽  
Test Function Method

We study the nonexistence of global solutions for new classes of nonlinear fractional differential inequalities. Namely, sufficient conditions are provided so that the considered problems admit no global solutions. The proofs of our results are based on the test function method and some integral estimates.

scholarly journals Nonexistence of Global Solutions to Higher-Order Time-Fractional Evolution Inequalities with Subcritical Degeneracy

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2765
Author(s):  
Ravi P. Agarwal ◽  
Soha Mohammad Alhumayan ◽  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Weak Solutions ◽  
Function Method ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Higher Order ◽  
Global Weak Solutions ◽  
Test Function ◽  
Test Function Method

In this paper, we study the nonexistence of global weak solutions to higher-order time-fractional evolution inequalities with subcritical degeneracy. Using the test function method and some integral estimates, we establish sufficient conditions depending on the parameters of the problems so that global weak solutions cannot exist globally.

scholarly journals Wave Breaking Phenomenon for DGH Equation with Strong Dissipation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Zhengguang Guo ◽  
Min Zhao
Keyword(s):  
Finite Time ◽  
Wave Breaking ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Dissipative Term

The present work is mainly concerned with the Dullin-Gottwald-Holm (DGH) equation with strong dissipative term. We establish some sufficient conditions to guarantee finite time blow-up of strong solutions.

The critical exponents of parabolic equations and blow-up in Rn

1998 ◽  
Vol 128 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Yuan-wei Qi
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Finite Time ◽  
Blow Up ◽  
Initial Value ◽  
The Cauchy Problem

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

A Modified Test Function Method for Damped Wave Equations

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Marcello D’Abbicco ◽  
Sandra Lucente
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Critical Exponent ◽  
Function Method ◽  
Wave Equations ◽  
Time Dependent ◽  
Small Data ◽  
Test Function ◽  
Test Function Method ◽  
Modified Test

AbstractIn this paper we use a modified test function method to derive nonexistence results for the semilinear wave equation with time-dependent speed and damping. The obtained critical exponent is the same exponent of some recent results on global existence of small data solutions.

scholarly journals Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2075
Author(s):  
Ratinan Boonklurb ◽  
Tawikan Treeyaprasert ◽  
Aong-art Wanna
Keyword(s):  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Initial Conditions ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Heat Equations ◽  
Quenching Rate ◽  
Linear Heat ◽  
Quenching Set

This article deals with finite-time quenching for the system of coupled semi-linear heat equations ut=uxx+f(v) and vt=vxx+g(u), for (x,t)∈(0,1)×(0,T), where f and g are given functions. The system has the homogeneous Neumann boundary conditions and the bounded nonnegative initial conditions that are compatible with the boundary conditions. The existence result is established by using the method of upper and lower solutions. We obtain sufficient conditions for finite time quenching of solutions. The quenching set is also provided. From the quenching set, it implies that the quenching solution has asymmetric profile. We prove the blow-up of time-derivatives when quenching occurs. We also find the criteria to identify simultaneous and non-simultaneous quenching of solutions. For non-simultaneous quenching, the corresponding quenching rate of solutions is given.

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