Radial symmetric solutions of the Cahn-Hilliard equation with degenerate mobility

In this paper we study the radial symmetric solutions of the two-dimensional Cahn-Hilliard equation with degenerate mobility. We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence and the nonnegativity of weak solutions

Early and late stage profiles for a new chemotaxis model with density-dependent jump probability and quorum-sensing mechanisms

In this paper, we derive a new chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, the new model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole body, we first estimate the expanding speed of tumour region as $O(t^{\beta})$ for $0<\beta<\frac{1}{2}$. Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state. The global convergence of the original weak solution to the steady state with exponential rate $O(e^{-ct})$ for some $c>0$ is also obtained

On critical Fujita exponents for the porous medium equation with a nonlinear boundary condition

AbstractWe establish the critical Fujita exponents for the solution of the porous medium equation ut=Δum, x∈R+N, t>0, subject to the nonlinear boundary condition −∂um/∂x1=up, x1=0, t>0, in multi-dimension

Multiplicity of positive solutions for a fourth-order quasilinear singular differential equation

This paper is concerned with the multiplicity of positive solutions of boundary value problem for the fourth-order quasilinear singular differential equation $$(|u''|^{p-2}u'')''=\lambda g(t)f(u),\quad 0<t<1,$$ where $p>1$, $\lambda>0$. We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive solutions. We have found a threshold $\lambda^*<+\infty$, such that if $0<\lambda\leq\lambda^*$, then the problem admits at least one positive solution; while if $\lambda \lambda^*$, then the problem has no positive solution. In particular, there exist at least two positive solutions for $0<\lambda<\lambda^*$