We consider a class of free boundary problems with Neumann boundary conditions. We would like to give certain results with regularity of solutions (mainly the local interior and boundary Lipschitz continuity). We will also show an explicit form of solution under well-specified conditions.

This paper is concerned with the asymptotic behavior of a quasilinear viscoelastic equation with nonlinear damping and memory. Assuming that the kernel $\mu (s)$ satisfies $$\mu'(s)\le -k_1\mu^m(s), \ 1\le m<\frac{3}{2}$$ we establish the exponential stability result for $m=1$ and the polynomial stability result for $1<m<\frac{3}{2}$.

In this paper, we are interested in a free boundary problem for a chemotaxis model with double free boundaries. We use contraction mapping principle and operator-theoretic approach to establish local solvability of a chemotaxis system in 1-Dimensional domain with non-constant coefficient free boundaries.

Option pricing is a major problem in quantitative finance. The Black-Scholes model proves to be an effective model for the pricing of options. In this paper a computational method known as the modified differential transform method has been employed to obtain the series solution of Black-Scholes equation with boundary conditions for European call and put options paying continuous dividends. The proposed method does not need discretization to find out the solution and thus the computational work is reduced considerably. The results are plotted graphically to establish the accuracy and efficacy of the proposed method.

We investigate a blowup problem of a reaction-advection-diffusion equation with double free boundaries and aim to use the dynamics of such a problem to describe the heat transfer and temperature change of a chemical reaction in advective environment with the free boundary representing the spreading front of the heat. We study the influence of the advection on the blowup properties of the solutions and conclude that large advection is not favorable for blowup. Moreover, we give the decay estimates of solutions and the two free boundaries converge to a finite limit for small initial data.

This paper considers the Cauchy problem of the semilinear wave equations with small initial data in the Schwarzschild space-time, $□_gu = |u_t | ^p ,$ where $g$ denotes the Schwarzschild metric. When $1< p<2$ and the initial data are supported far away from the black hole, we can prove that the lifespan of the spherically symmetric solution obtains the same order as the semilinear wave equation evolving in the Minkowski space-time by introducing an auxiliary function.

Let $W^{1,n} (\mathbb{R}^n)$ be the standard Sobolev space. For any $\tau>0$ and $p>n>2,$ we denote $$\lambda_{n,p}=\inf\limits_{u\in W^{1,n}(\mathbb{R}^n),\ u\not\equiv 0}\frac{\int_{\mathbb{R}^n}(|∇u|^n+\tau|u|^n)dx}{(\int_{\mathbb{R}^n}|u|^p dx)^{\frac{n}{p}}}. $$ Define a norm in $W^{1,n} (\mathbb{R}^n)$ by $$||u||_{n,p}=(\int_{\mathbb{R}^n}(|∇u|^n+\tau|u|^n)dx-\alpha(\int_{\mathbb{R}^n}|u|^pdx)^{\frac{n}{p}})^{\frac{1}{n}},$$ where $0 ≤ α < λ_{n,p}.$ Using a rearrangement argument and blow-up analysis, we will prove $$\sup\limits_{u\in W^{1,n}(\mathbb{R}^n), \ ||u||_{n,p}\le 1}\int_{\mathbb{R}^n}\left(e^{\alpha_n |u|^{\frac{n}{n-1}}}-\sum^{n-1}_{j=0}\frac{\alpha_n^j|u|^{\frac{jn}{n-1}}}{j!} \right)dx$$ can be attained by some function $u_0 ∈ W^{1,n} (\mathbb{R}^n )∩C^1 (\mathbb{R}^n)$ with $∥u_0∥_{n,p}= 1,$ here $α_n = nω ^{\frac{1}{ n−1}}_{n−1}$ and $ω_{n−1}$ is the measure of the unit sphere in $\mathbb{R}^n .$

We study local dynamics of a diffusive predator-prey model in a spatially heterogeneous environment, where intrinsic growth rate of the prey is spatially homogeneous, whereas carrying capacity of the habitat is spatially inhomogeneous. In comparison with the existing predator-prey models, the stability of semi-trivial steady state of this model displays distinct properties. For example, for certain intermediate ranges of the death rate of the predator, the semi-trivial steady state can change its stability at least once as the dispersal rate of the prey varies from small to large, while the stability of the semi-trivial steady state is immune from the dispersal rate of the predator.