scholarly journals Nonexistence result for the generalized Tricomi equation with the scale-invariant damping, mass term and time derivative nonlinearity

2021 ◽  
pp. 1-21
Author(s):  
Moahmed Fahmi Ben Hassen ◽  
Makram Hamouda ◽  
Mohamed Ali Hamza ◽  
Hanen Khaled Teka
Keyword(s):  
Blow Up ◽  
Time Derivative ◽  
Mass Term ◽  
Scale Invariant ◽  
Small Initial Data ◽  
Damping Coefficients ◽  
Nonexistence Result ◽  
Tricomi Equation ◽  
Speed Of Propagation ◽  
Generalized Tricomi Equation

In this article, we consider the damped wave equation in the scale-invariant case with time-dependent speed of propagation, mass term and time derivative nonlinearity. More precisely, we study the blow-up of the solutions to the following equation: ( E ) u t t − t 2 m Δ u + μ t u t + ν 2 t 2 u = | u t | p , in  R N × [ 1 , ∞ ) , that we associate with small initial data. Assuming some assumptions on the mass and damping coefficients, ν and μ > 0, respectively, we prove that blow-up region and the lifespan bound of the solution of ( E ) remain the same as the ones obtained for the case without mass, i.e. ( E ) with ν = 0 which constitutes itself a shift of the dimension N by μ 1 + m compared to the problem without damping and mass. Finally, we think that the new bound for p is a serious candidate to the critical exponent which characterizes the threshold between the blow-up and the global existence regions.

scholarly journals Blow-up of energy solutions for the semilinear generalized Tricomi equation with nonlinear memory term

2021 ◽  
Vol 6 (10) ◽  
pp. 10907-10919
Author(s):  
Jincheng Shi ◽  
◽  
Jianye Xia ◽  
Wenjing Zhi ◽  
◽  
...  
Keyword(s):  
Relaxation Function ◽  
Blow Up ◽  
Combined Effect ◽  
Time Dependent ◽  
Memory Term ◽  
Nonlinear Memory ◽  
Tricomi Equation ◽  
Generalized Tricomi Equation

<abstract><p>In this paper, we investigate blow-up conditions for the semilinear generalized Tricomi equation with a general nonlinear memory term in $ \mathbb{R}^n $ by using suitable functionals and employing iteration procedures. Particularly, a new combined effect from the relaxation function and the time-dependent coefficient is found.</p></abstract>

scholarly journals Nonexistence of Global Solutions to A Semilinear Wave Equation with Scale Invariant Damping

2021 ◽  
pp. 10-26
Author(s):  
Changwang Xiao
Keyword(s):  
Wave Equation ◽  
Critical Case ◽  
Global Solutions ◽  
Fundamental Solutions ◽  
Scale Invariant ◽  
Semilinear Wave Equation ◽  
Change Of Variables ◽  
Tricomi Equation ◽  
Blowup Result ◽  
Generalized Tricomi Equation

We obtain a blowup result for solutions to a semilinear wave equation with scale-invariant dissipation. We perform a change of variables that transforms our starting equation into a Generalized Tricomi equation, then apply Kato’s lemma, we can prove a blowup result for solutions to the transformed equation under some assumptions on the initial data. In the critical case, we use the fundamental solutions of the Generalized Tricomi equation to modify Kato’s lemma to deal with it.

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.

scholarly journals Existence and nonexistence of entire solutions to the logistic differential equation

2003 ◽  
Vol 2003 (17) ◽  
pp. 995-1003 ◽  
Author(s):  
Marius Ghergu ◽  
Vicentiu Radulescu
Keyword(s):  
Blow Up ◽  
Entire Solutions ◽  
One Dimensional ◽  
Nonexistence Result ◽  
Nonnegative Solutions ◽  
Existence And Nonexistence ◽  
Logistic Problem ◽  
Whole Space ◽  
The Mean ◽  
The One

We consider the one-dimensional logistic problem(rαA(|u′|)u′)′=rαp(r)f(u)on(0,∞),u(0)>0,u′(0)=0, whereαis a positive constant andAis a continuous function such that the mappingtA(|t|)is increasing on(0,∞). The framework includes the case wherefandpare continuous and positive on(0,∞),f(0)=0, andfis nondecreasing. Our first purpose is to establish a general nonexistence result for this problem. Then we consider the case of solutions that blow up at infinity and we prove several existence and nonexistence results depending on the growth ofpandA. As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative solutions, excepting the trivial one.

scholarly journals A nonlocal problem for a generalized Trikomi equation with a spectral parameter in the unbounded domain

2021 ◽  
pp. 17-26
Author(s):  
Р.Т. Зуннунов
Keyword(s):  
Half Plane ◽  
Spectral Parameter ◽  
Unbounded Domain ◽  
Nonlocal Problem ◽  
Existence Of A Solution ◽  
Tricomi Equation ◽  
Energy Integrals ◽  
Uniqueness Of The Solution ◽  
Upper Half Plane ◽  
Generalized Tricomi Equation

В данной статье изучена нелокальная задача для обобщенного уравнения Трикоми со спектральным параметром в неограниченной области эллиптическая часть которой является верхней полуплоскостью. Единственность решения поставленной задачи доказана методом интегралов энергии. Существование решения поставленной задачи доказана методом функций Грина и интегральных уравнений. In this article, we study a nonlocal problem for the generalized Tricomi equation with a spectral parameter in an unbounded domain, the elliptic part of which is the upper half-plane. The uniqueness of the solution to the problem posed is proved by the method of energy integrals. The existence of a solution to the problem is proved by the method of Green’s functions and integral equations.

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