Self-Similar Solutions of a Convection Diffusion Equation And Related Semilinear Elliptic Problems

1990 ◽  
Vol 15 (2) ◽  
pp. 139-157 ◽  
Author(s):  
J. Aguirre ◽  
M. Escobedo ◽  
E. Zuazua
Keyword(s):  
Diffusion Equation ◽  
Elliptic Problems ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Self Similar ◽  
Similar Solutions

Asymptotically Self-Similar Solution for the Convection-Diffusion Equation

2014 ◽  
Vol 623 ◽  
pp. 97-100 ◽  
Author(s):  
Xue Yu ◽  
Qiu Sheng Zhang
Keyword(s):  
Diffusion Equation ◽  
Global Solutions ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Self Similar Solution ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Diffusion And Convection ◽  
Physical Phenomena ◽  
Physical Quantities

We study the Cauchy problem for the convection-diffusion equation, which describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to diffusion and convection processes. For, we show the continuous dependence upon the initial data. Moreover, asymptotically self–similar global solutions are investigated with nonhomogeneous initial date.

scholarly journals Eternal solutions for a reaction-diffusion equation with weighted reaction

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Razvan Gabriel Iagar ◽  
Ariel Sánchez
Keyword(s):  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Growing Up ◽  
Convection Diffusion ◽  
Eternal Solutions ◽  
Self Similar ◽  
Similar Solutions

<p style='text-indent:20px;'>We prove existence and uniqueness of <i>eternal solutions</i> in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>posed in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ m&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0&lt;p&lt;1 $\end{document}</tex-math></inline-formula> and the critical value for the weight</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \sigma = \frac{2(1-p)}{m-1}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Existence and uniqueness of some specific solution holds true when <inline-formula><tex-math id="M4">\begin{document}$ m+p\geq2 $\end{document}</tex-math></inline-formula>. On the contrary, no eternal solution exists if <inline-formula><tex-math id="M5">\begin{document}$ m+p&lt;2 $\end{document}</tex-math></inline-formula>. We also classify exponential self-similar solutions with a different interface behavior when <inline-formula><tex-math id="M6">\begin{document}$ m+p&gt;2 $\end{document}</tex-math></inline-formula>. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.</p>

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