scholarly journals Asymptotic Behavior of Solutions to Fast Diffusive Non-Newtonian Filtration Equations Coupled by Nonlinear Boundary Sources

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Wang Zejia ◽  
Wang Shunti ◽  
Zhang Chengbin
Keyword(s):  
Asymptotic Behavior ◽  
Global Existence ◽  
Nonlinear Boundary ◽  
Large Time ◽  
Large Time Behavior ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Multidimensional Problem ◽  
Critical Curves

This paper is concerning the asymptotic behavior of solutions to the fast diffusive non-Newtonian filtration equations coupled by the nonlinear boundary sources. We are interested in the critical global existence curve and the critical Fujita curve, which are used to describe the large-time behavior of solutions. It is shown that the above two critical curves are both the same for the multidimensional problem we considered.

scholarly journals Asymptotic Behavior of Solutions to Reaction-Diffusion Equations with Dynamic Boundary Conditions and Irregular Data

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Yonghong Duan ◽  
Chunlei Hu ◽  
Xiaojuan Chai
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Conditions ◽  
Reaction Diffusion ◽  
Large Time Behavior ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Dynamic Boundary

This paper is concerned with the asymptotic behavior of solutions to reaction-diffusion equations with dynamic boundary conditions as well as L1-initial data and forcing terms. We first prove the existence and uniqueness of an entropy solution by smoothing approximations. Then we consider the large-time behavior of the solution. The existence of a global attractor for the solution semigroup is obtained in L1(Ω¯,dν). This extends the corresponding results in the literatures.

scholarly journals Large Time Behavior of Solutions to the Nonlinear Heat Equation with Absorption with Highly Singular Antisymmetric Initial Values

2020 ◽  
Vol 20 (2) ◽  
pp. 311-337
Author(s):  
Hattab Mouajria ◽  
Slim Tayachi ◽  
Fred B. Weissler
Keyword(s):  
Heat Equation ◽  
Large Time ◽  
Large Time Behavior ◽  
Nonlinear Heat Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Time Behavior ◽  
Initial Values ◽  
Self Similar ◽  
Similar Solutions

AbstractIn this paper, we study global well-posedness and long-time asymptotic behavior of solutions to the nonlinear heat equation with absorption, {u_{t}-\Delta u+\lvert u\rvert^{\alpha}u=0}, where {u=u(t,x)\in\mathbb{R}}, {(t,x)\in(0,\infty)\times\mathbb{R}^{N}} and {\alpha>0}. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables {x_{1},x_{2},\ldots,x_{m}} for some {m\in\{1,2,\ldots,N\}}, such as {u_{0}=(-1)^{m}\partial_{1}\partial_{2}\cdots\partial_{m}\lvert\,{\cdot}\,% \rvert^{-\gamma}\in\mathcal{S}^{\prime}(\mathbb{R}^{N})}, {0<\gamma<N}. In fact, we show global well-posedness for initial data bounded in an appropriate sense by {u_{0}} for any {\alpha>0}. Our approach is to study well-posedness and large time behavior on sectorial domains of the form {\Omega_{m}=\{x\in\mathbb{R}^{N}:x_{1},\ldots,x_{m}>0\}}, and then to extend the results by reflection to solutions on {\mathbb{R}^{N}} which are antisymmetric. We show that the large time behavior depends on the relationship between α and {\frac{2}{\gamma+m}}, and we consider all three cases, α equal to, greater than, and less than {\frac{2}{\gamma+m}}. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.

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