Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation

Abstract

In this paper, we investigate the positive solution of nonlinear degenerate equation u t = f (u)( u + a(x) Ω u dx) with Dirichlet boundary condition. The blow-up criteria is obtained. Furthermore, we prove that under certain conditions, the solutions have global blow-up. When f (u) = u p , 0 < p 1, we gained blow-up rate estimate.

Global Existence and Blow-Up for a Class of Degenerate Parabolic Systems with Localized Source

“…It also arises in the study of the flow of a fluid through a homogeneous isotropic rigid porous medium with internal localized source, see [3,[6][7][8]. The studies on the blow-up and global existence of solutions for the equations or systems not in divergence form with local, nonlocal or localized terms can be found in [2,4,5,[9][10][11][12][16][17][18][19][20][21].…”

“…have been studied in [4] and [9]. When a is a positive constant, Deng et al in [9] proved that the solution blows up in finite time if and only if ∞ δ ds sf (s) < ∞ and a > 1/μ, where δ > 0 is a constant and μ is defined as…”

Some Types of Reaction-Diffusion Systems With Nonlocal Boundary Conditions

“…It was shown that the blow-up occurs for large initial data if q > p > 1, while all solutions exist globally if 1 ≤ q < p. As for the critical case p = q, they proved that whether or not the solutions blow up in finite time depending on the comparison of |Ω| and k. Global blow-up property was also proved. For more works on parabolic equations or systems with localized or nonlocal sources, we refer the readers to [1,6,18,25,29]. On the other hand, parabolic equations with nonlocal boundary conditions also arise naturally in applied sciences (see [4,9,10]).…”

Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system

“…Such a problem can describe a variety of physical phenomena which arise, for example, in the study of the flow of a fluid through a homogeneous isotropic rigid porous medium or in the studies of population dynamics (see [1], [3], [5], [6], [7] and references therein).…”

“…with the homogeneous Dirichlet boundary condition and positive initial data, has been studied in [3], [4], where p > 0. It was proved that the solution blows up in finite time if and only if µ > 1/a, where µ is defined in (1.4) below.…”