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 Blow-Up of Solutions for a Class of Nonlinear Pseudoparabolic Equations with a Memory Term

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Huafei Di ◽  
Yadong Shang
Keyword(s):  
Boundary Condition ◽  
Relaxation Function ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Initial Energy ◽  
Memory Term ◽  
Initial Condition ◽  
Pseudoparabolic Equation ◽  
Pseudoparabolic Equations ◽  
Nonlinear Pseudoparabolic Equation

We consider the nonlinear pseudoparabolic equation with a memory termut-Δu-Δut+∫0tλt-τΔuτdτ=div∇up-2u+u1+α,x∈Ω,t>0, with an initial condition and Dirichlet boundary condition. Under negative initial energy and suitable conditions onp,α, and the relaxation functionλ(t), we prove a finite-time blow-up result by using the concavity method.

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 analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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