scholarly journals Blow-up rates and uniqueness of entire large solutions to a semilinear elliptic equation with nonlinear convection term

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Bo Li ◽  
Haitao Wan
Keyword(s):  
Elliptic Equation ◽  
Blow Up ◽  
Semilinear Elliptic Equation ◽  
Convection Term ◽  
Nonlinear Convection ◽  
Large Solutions ◽  
Semilinear Elliptic ◽  
Nonlinear Convection Term

Morse index and symmetry breaking for an elliptic equation with negative exponent in expanding annuli

2017 ◽  
Vol 147 (6) ◽  
pp. 1215-1232
Author(s):  
Zongming Guo ◽  
Linfeng Mei ◽  
Zhitao Zhang
Keyword(s):  
Elliptic Equation ◽  
Symmetry Breaking ◽  
Morse Index ◽  
Blow Up ◽  
Semilinear Elliptic Equation ◽  
Radial Solutions ◽  
Entire Solutions ◽  
Semilinear Elliptic ◽  
Image Position

Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with negative exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equationThe main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equationwith N ⩾ 2 and p > 0.

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.

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