Asymptotic profile of solutions for the damped wave equation with a nonlinear convection term

2017 ◽  
Vol 40 (18) ◽  
pp. 7760-7779 ◽  
Author(s):  
Masakazu Kato ◽  
Yoshihiro Ueda
Keyword(s):  
Wave Equation ◽  
Convection Term ◽  
Damped Wave Equation ◽  
Nonlinear Convection ◽  
Asymptotic Profile ◽  
Nonlinear Convection Term

scholarly journals Blow-up for semidiscrete forms of some nonlinear parabolic equations with convection

2020 ◽  
pp. 267-288
Author(s):  
N'Guessan Koffi ◽  
Diabate Nabongo ◽  
Toure Kidjegbo Augustin
Keyword(s):  
Parabolic Equations ◽  
Numerical Approximation ◽  
Blow Up ◽  
Mesh Size ◽  
Nondecreasing Function ◽  
Nonlinear Parabolic Equations ◽  
Convection Term ◽  
Nonlinear Convection ◽  
Nonlinear Parabolic ◽  
Nonlinear Convection Term

This paper concerns the study of the numerical approximation for the following parabolic equations with a nonlinear convection term $$\\ \left\{% \begin{array}{ll} \hbox{$u_t(x,t)=u_{xx}(x,t)-g(u(x,t))u_{x}(x,t)+f(u(x,t)),\quad 0<x<1,\; t>0$,} \\ \hbox{$u_{x}(0,t)=0, \quad u_{x}(1,t)=0,\quad t>0$,} \\ \hbox{$u(x,0)=u_{0}(x) > 0,\quad 0\leq x \leq 1$,} \\ \end{array}% \right. $$ \newline where $f:[0,+\infty)\rightarrow [0,+\infty)$ is $C^3$ convex, nondecreasing function,\\ $g:[0,+\infty)\rightarrow [0,+\infty)$ is $C^1$ convex, nondecreasing function,\newline $\displaystyle\lim_{s\rightarrow +\infty}f(s)=+\infty$, $\displaystyle\lim_{s\rightarrow +\infty}g(s)=+\infty$, $\displaystyle\lim_{s\rightarrow +\infty}\frac{f(s)}{g(s)}=+\infty$\newline and $\displaystyle\int^{+\infty}_{c}\frac{ds}{f(s)}<+\infty$ for $c>0$. We obtain some conditions under which the solution of the semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove that the semidiscrete blow-up time converges to the real one, when the mesh size goes to zero. Finally, we give some numerical results to illustrate ours analysis.

Asymptotic Behavior of The Unique Solution to a Singular Elliptic Problem With Nonlinear Convection Term And Singular Weight

2008 ◽  
Vol 8 (2) ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Local Solution ◽  
Variation Theory ◽  
Convection Term ◽  
Unique Classical Solution ◽  
Nonlinear Convection ◽  
Comparison Functions ◽  
Exact Asymptotic Behavior ◽  
Nonlinear Convection Term ◽  
Singular Dirichlet Problem

AbstractBy Karamata regular variation theory, we first derived the exact asymptotic behavior of the local solution to the problem -φʹʹ(s) = g(φ(s)), φ(s) > 0, s ∈ (0, a) and φ(0) = 0. Then, by a perturbation method and constructing comparison functions, we derived the exact asymptotic behavior of the unique classical solution near the boundary to a singular Dirichlet problem -Δu = b(x)g(u) + λ|▽u|

scholarly journals Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term

2015 ◽  
Vol 8 (6) ◽  
pp. 1331-1339 ◽  
Author(s):  
María Rosa ◽  
◽  
María de los Santos Bruzón ◽  
María de la Luz Gandarias
Keyword(s):  
Conservation Laws ◽  
Lie Symmetries ◽  
Fisher Equation ◽  
Convection Term ◽  
Nonlinear Convection ◽  
Nonlinear Convection Term

A note on the breaking of waves

1968 ◽  
Vol 303 (1475) ◽  
pp. 493-496 ◽  
Keyword(s):  
Water Waves ◽  
Model Equation ◽  
Maximum Height ◽  
Periodic Waves ◽  
Square Root ◽  
Convection Term ◽  
Nonlinear Convection ◽  
Dispersion Effects ◽  
Present Argument ◽  
Nonlinear Convection Term

A model equation for water waves has been suggested by Whitham (1967) to study, qualitatively at least, the different kinds of breaking. This is an integro-differential equation which combines a typical nonlinear convection term with an integral for the dispersive effects and it is of independent mathematical interest. For an approximate kernel of the form e - b | x | , it is shown first that solitary or periodic waves have a maximum height with sharp crests and secondly that waves which are sufficiently asymmetric break into ‘bores’. The second part applies to a wide class of bounded kernels, but the kernel giving the correct dispersion effects of water waves has a square root infinity and the present argument does not go through. Nevertheless, the possibility of the two kinds of breaking in such integro-differential equations is demonstrated.

scholarly journals On a Singular Robin Problem with Convection Terms

2020 ◽  
Vol 20 (4) ◽  
pp. 895-909 ◽  
Author(s):  
Umberto Guarnotta ◽  
Salvatore A. Marano ◽  
Dumitru Motreanu
Keyword(s):  
Regularity Theory ◽  
Uniqueness Result ◽  
Robin Problem ◽  
Convection Term ◽  
Nonlinear Regularity ◽  
Nonlinear Convection ◽  
Nonlinear Regularity Theory ◽  
Truncation Techniques ◽  
Robin Boundary ◽  
Nonlinear Convection Term

AbstractIn this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly exploit sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity theory, as well as fixed point arguments. A uniqueness result is also presented.

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