21 research outputs found
Blow-up problems for quasilinear reaction diffusion equations with weighted nonlocal source
In this paper, we investigate the following quasilinear reaction diffusion equations
Here is a bounded domain in with smooth boundary . Weighted nonlocal source satisfies
where are some positive constants and 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
Here is a bounded domain in with smooth boundary . Weighted nonlocal source satisfies
where are some positive constants and 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 and blow-up solutions for nonlinear parabolic problems with a gradient term under Robin boundary conditions
Blow-up solutions, global existence, and exponential decay estimates for second order parabolic problems
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 , the spatial region D in RN ( N≥2 ) 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