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 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.

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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