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This article studies the asymptotic analysis of a coupled model arising in radiative transfer in $ \mathbb{T}^{3} $ with one-speed velocity. The model consists of a kinetic equation satisfied by the specific intensity of radiation coupled to a nonhomogeneous heat equation satisfied by the material temperature. We show that the radiative intensity can be approximated by the sum of the interior solution and initial layer, while the temperature can be approximated by the interior solution for the general initial data. [ABSTRACT FROM AUTHOR]

We consider the diffusion asymptotics of a coupled model arising in radiative transfer in a unit disk in R 2 with one-speed velocity. The model consists of a kinetic equation satisfied by the specific intensity of radiation coupled a nonhomogeneous heat equation satisfied by the material temperature. For the well-prepared data, we prove the existence and the nonequilibrium diffusion limit of solutions to the initial boundary problem for the coupled model. [ABSTRACT FROM AUTHOR]

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term: ut = uxx + xm|ux|p, t > 0, 0 < x < 1,where p > m + 2, m ≥ 0. Zhang and Hu [Discrete Contin. Dyn. Syst. 26 (2010) 767-779] showed that finite time gradient blowup occurs at the boundary and the accurate blowup rate is also obtained for super-critical boundary value. Throughout this paper, we present a complete large time behavior of a classical solution : is global and converges to the unique stationary solution in norm for subcritical boundary value, and blows up in infinite time for critical boundary value. Gradient growup rate is also established by the method of matched asymptotic expansions. In addition, gradient estimate of solutions is obtained by the Bernstein-type arguments. [ABSTRACT FROM AUTHOR]

In this paper we consider a free boundary tumor model with angiogenesis. The model consists of a reaction-diffusion equation describing the concentration of nutrients σσ and an elliptic equation describing the distribution of the internal pressure p. The vasculature supplies nutrients to the tumor, so that ∂σ/∂n + β(σ − σ) = 0 holds on the boundary, where a positive constant β is the rate of nutrient supply to the tumor and ˉσ is the nutrient concentration outside the tumor. The tumor cells proliferate at a rate μ. If 0 < ˜σ < σ, where ˜σ is a threshold concentration for proliferating, then there exists a unique radially symmetric stationary solution (σS(r),pS(r),RS). In this paper, we found a function μ∗ = μ∗ (RS) such that if μ < μ∗ then the radially symmetric stationary solution is linearly stable with respect to non-radially symmetric perturbations, whereas if μ > μ∗ then the radially symmetric stationary solution is linearly unstable. [ABSTRACT FROM AUTHOR]

This paper is focused on the gradient blowup rate for the following general semilinear parabolic equation withbeing a smoothly bounded domain inandbeing a smooth function. Under some suitable assumptions on, we establish the gradient blowup rate lower bound by the rescaling method for any. While the upper bound which is established by the comparison method can be obtained only for. At last, we provide some specific examples to verify our results. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we consider nonnegative solutions of the quasilinear parabolic equation with -Laplace operator , where and . Our main result is that there is no nontrivial positive bounded radial entire solution. The proof is based on intersection comparison arguments, which can be viewed as a sophisticated form of the maximum principle and has been used to deal with the semilinear heat equation by Poláčik and Quittner [Peter Poláčik, Pavol Quittner, A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation, Nonlinear Analysis TMA 64 (2006) 1679–1689] and the porous medium equation by Souplet [Ph. Souplet, An optimal Liouville-type theorem for radial entire solutions of the porous medium equation with source, J. Differential Equations 246 (2009) 3980–4005]. [Copyright &y& Elsevier]