scholarly journals A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yang Liu ◽  
Wenke Li
Keyword(s):  
Thin Films ◽  
Epitaxial Growth ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic Equations ◽  
Potential Wells ◽  
Nonlinear Parabolic

<p style='text-indent:20px;'>In this paper, the initial-boundary value problem for a class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films is studied. By means of the theory of potential wells, the global existence, asymptotic behavior and finite time blow-up of weak solutions are obtained.</p>

XXII.—A Priori Estimates and Nonlinear Parabolic Equations of Arbitrary Order

1972 ◽  
Vol 69 (4) ◽  
pp. 287-293
Author(s):  
D. E. Edmunds ◽  
C. A. Stuart
Keyword(s):  
Parabolic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic Equations ◽  
Linear Parabolic Equation ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Initial Boundary Value

SynopsisIn this paper it is shown that the question of the existence of a classical solution of the first initial-boundary value problem for a non-linear parabolic equation may be reduced to the problem of the derivation of suitable a priori bounds.

scholarly journals A nonlinear parabolic equation modelling surfactant diffusion

2000 ◽  
Vol 11 (4) ◽  
pp. 413-432
Author(s):  
XINFU CHEN ◽  
CHAOCHENG HUANG ◽  
JENNIFER ZHAO
Keyword(s):  
Parabolic Equations ◽  
Single Point ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic Equations ◽  
Well Posedness ◽  
Initial Value ◽  
Equation Modelling ◽  
Nonlinear Parabolic ◽  
Initial Boundary Value

An initial-boundary value problem for nonlinear parabolic equations modelling surfactant diffusions is investigated. The boundary conditions are of nonlinear adsorptive types, and the initial value has a single point jump. We study the well-posedness of the problem, the convergence of a numerical scheme, and the regularity as well as quantitative behaviour of solutions.

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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