Blow-up problems for quasilinear reaction diffusion equations with weighted nonlocal source

In this paper, we investigate the following quasilinear reaction diffusion equations $$\begin{cases} \left(b(u)\right)_t =\nabla\cdot\left(\rho\left(|\nabla u|^2\right)\nabla u\right)+c(x)f(u) &\hbox{ in } \Omega\times(0,t^{*}),\\ \frac{\partial u}{\partial \nu}=0 &\hbox{ on } \partial\Omega\times(0,t^{*}), \\ u(x,0)=u_{0}(x)\geq0 & \hbox{ in } \overline{\Omega}. \end{cases}$$ Here $\Omega$ is a bounded domain in $\mathbb{R}^{n}\ (n\geq2)$ with smooth boundary $\partial\Omega$. Weighted nonlocal source satisfies $$c(x)f(u(x,t))\leq a_1+a_2\left(u(x,t)\right)^{p}\left(\int_{\Omega}\left(u(x,t)\right)^{\alpha}{\rm d}x\right)^{m},$$ where $a_2,p,\alpha$ are some positive constants and $a_1, m$ are some nonnegative constants. We make use of a differential inequality technique and Sobolev inequality to obtain a lower bound for the blow-up time of the solution. In addition, an upper bound for the blow-up time is also derived

Blow-up problems for quasilinear reaction diffusion equations with weighted nonlocal source

In this paper, we investigate the following quasilinear reaction diffusion equations $$\begin{cases} \left(b(u)\right)_t =\nabla\cdot\left(\rho\left(|\nabla u|^2\right)\nabla u\right)+c(x)f(u) &\hbox{ in } \Omega\times(0,t^{*}),\\ \frac{\partial u}{\partial \nu}=0 &\hbox{ on } \partial\Omega\times(0,t^{*}), \\ u(x,0)=u_{0}(x)\geq0 & \hbox{ in } \overline{\Omega}. \end{cases}$$ Here $\Omega$ is a bounded domain in $\mathbb{R}^{n}\ (n\geq2)$ with smooth boundary $\partial\Omega$. Weighted nonlocal source satisfies $$c(x)f(u(x,t))\leq a_1+a_2\left(u(x,t)\right)^{p}\left(\int_{\Omega}\left(u(x,t)\right)^{\alpha}{\rm d}x\right)^{m},$$ where $a_2,p,\alpha$ are some positive constants and $a_1, m$ are some nonnegative constants. We make use of a differential inequality technique and Sobolev inequality to obtain a lower bound for the blow-up time of the solution. In addition, an upper bound for the blow-up time is also derived

Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions

Abstract This paper is devoted to studying the global existence and blow-up results for the following p-Laplacian parabolic problems: {(h(u))t=∇⋅(|∇u|p−2∇u)+f(u)in D×(0,t∗),∂u∂n=g(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾. \textstyle\begin{cases} (h(u) )_{t} =\nabla\cdot (|\nabla u|^{p-2}\nabla u )+f(u) &\mbox{in } D\times(0,t^{*}), \\ \frac{\partial u}{\partial n}=g(u) &\mbox{on } \partial D\times (0,t^{*}), \\ u(x,0)=u_{0}(x)\geq0 & \mbox{in } \overline{D}. \end{cases} Here p>2 $p>2$, the spatial region D in RN $\mathbb{R}^{N}$ ( N≥2 $N\geq2$) is bounded, and ∂D is smooth. We set up conditions to ensure that the solution must be a global solution or blows up in some finite time. Moreover, we dedicate upper estimates of the global solution and the blow-up rate. An upper bound for the blow-up time is also specified. Our research relies mainly on constructing some auxiliary functions and using the parabolic maximum principles and the differential inequality technique

Blow-Up Solutions and Global Existence for Quasilinear Parabolic Problems with Robin Boundary Conditions

We study the blow-up and global solutions for a class of quasilinear parabolic problems with Robin boundary conditions. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of blow-up solution, an upper bound for the “blow-up time,” an upper estimate of the “blow-up rate,” the sufficient conditions for the existence of global solution, and an upper estimate of the global solution are specified

Blow-Up Phenomena for Nonlinear Reaction-Diffusion Equations under Nonlinear Boundary Conditions

This paper deals with blow-up and global solutions of the following nonlinear reaction-diffusion equations under nonlinear boundary conditions: g(u)t=∇·au∇u+fu  in  Ω×0,T,  ∂u/∂n=bx,u,t  on  ∂Ω×(0,T),  u(x,0)=u0(x)>0,  in  Ω¯, where Ω⊂RN  (N≥2) is a bounded domain with smooth boundary ∂Ω. We obtain the conditions under which the solutions either exist globally or blow up in a finite time by constructing auxiliary functions and using maximum principles. Moreover, the upper estimates of the “blow-up time,” the “blow-up rate,” and the global solutions are also given