In this paper we investigate questions of existence and uniqueness of equilibrium solutions of the inhomogeneous Bénard equations with exterior force $f$ which may be generated by a magnetic field which is not too strong. Two types of results are obtained. The first, based on a priori estimates and degree arguments states that there is a certain range $\lambda\leq \lambda_c+\delta_1$ for the Rayleigh parameter $\lambda$, for which existence can be asserted for any given force $f$, while the second result says that for $\lambda <\lambda_c$, $\lambda_c-\lambda$ small, there are forces $f$ which give rise to three different solutions. Here, $\lambda^2_c=R_c$ is the critical Rayleigh number which enters basically into our considerations.

We study large time asymptotic behavior of small solutions to the quadratic nonlinear Schrödinger equation \begin{equation} \begin{cases} i\partial _{t}u+\frac{1}{2}\Delta u=\lambda u^{2}+\mu \overline{u}^{2},\text{ }(t,x)\in {\mathbf{R}}\times {\mathbf{R}}^{3}, \\ u(0,x)=u_{0}(x),\text{ }x\in {\mathbf{R}}^{3}, \end{cases} \tag*{(0.1)} \end{equation} where $\lambda$, $\mu\in\mathbf{C}$. We prove the $\mathbf{L}^{p}(2\leq p\leq \infty )$ - time decay estimate of solutions \begin{equation*} \big\| u ( t ) \big\| _{\mathbf{L}^{p}}\leq C\big( 1+ | t | \big) ^{-\frac{3}{2}\big( 1-\frac{2}{p}\big) }\big( \left\| u_{0}\right\| _{3,0}+\left\| u_{0}\right\| _{1,2}\big) \end{equation*} under the condition that $u_{0}\in \mathbf{H}^{3,0}\cap \mathbf{H}^{1,2}$ is sufficiently small, where \begin{equation*} \mathbf{H}^{m,k}=\left\{ \phi \in \mathbf{L}^{2}:\big\| \phi \big\| _{m,k}\equiv \big\| \langle x\rangle ^{k}\langle i\nabla \rangle ^{m}\phi \big\| _{\mathbf{L}^{2}} <\infty \right\} \end{equation*} is the weighted Sobolev space and $\langle x\rangle =\sqrt{1+x^{2}}$.

In this paper we will show that if $0\le u_0\in L^1(R^2)\cap L^p(R^2)$ for some constant $p>1$ and $f\in C([0,\infty))$ with $f\ge 2$ on $[0,\infty)$, then the solution of the equation $u_t=\Delta$ log $u$, $u>0$, in $R^2\times (0,T)$, $u(x,0)=u_0(x)$ on $R^2$, which satisfies the conditions $\int_{R^2}u(x,t)dx=\int_{R^2}u_0(x)dx-2\pi\int_0^tf(s)ds$ for all $0\le t\le T$ and log $u/$log $|x|\to -f$ uniformly as $|x|\to\infty$ on any compact subset of $(0,T)$ is unique where $T$ is given by $\int_{R^2}u_0(x)dx=2\pi\int_0^Tf(s)ds$. For $0\le u_0(x)\le C\min (1,(|x|\text{log }|x|)^{-2})$ or $u_0\in L^1(R^2)\cap L^p(R^2)$ for some $p>1$ with compact support and $f\in C([0,\infty))$, $f\ge 2$, we prove existence of solution satisfying the decay condition log $u/$log $|x|\to -f$ uniformly as $|x|\to\infty$ on any compact subset of $(0,T)$. We will also show that any solution obtained as the limit of solutions of certain Dirichlet problems in bounded domains will satisfy the same decay condition and is hence unique.

Mathematical models of many pattern formation problems in biology are reaction-diffusion systems. These systems are important for computer simulations of the patterns, parameter estimations as well as analysis of the biological properties. In order to solve reaction-diffusion systems efficiently, fast and stable numerical algorithms are essential for the pattern formation problems. In this paper, a fairly general reaction-diffusion system is considered. We propose a fully implicit discretization combined with a multigrid V-cycle solver for solving the reaction-diffusion system. Theorems about unconditional stability and convergence of the algorithm are given to show that the algorithm is highly stable and efficient. Numerical experiment results are given for two reaction-diffusion systems that can be used for generating animal coat markings. We also show the comparison results of the multigrid algorithm with other numerical algorithms.

Let $(1) Rh=f$, $0\leq x\leq L$, $Rh=\int^L_0 R(x,y)h(y)\,dy$, where the kernel $R(x,y)$ satisfies the equation $QR=P\delta(x-y)$. Here, $Q$ and $P$ are formal differential operators of order $n$ and $m <n$, respectively, $n$ and $m$ are nonnegative even integers, $n>0$, $m\geq 0$, $Qu:=q_n(x)u^{(n)} + \sum^{n-1}_{j=0} q_j(x) u^{(j)}$, $Ph:=h^{(m)} +\sum^{m-1}_{j=0} p_j(x) h^{(j)}$, $q_n(x)\geq c>0$, the coefficients $q_j(x)$ and $p_j(x)$ are smooth functions defined on $\mathbb R$, $\delta(x)$ is the delta-function, $f\in H^\alpha(0,L)$, $\alpha:=\frac{n-m}{2}$, $H^\alpha$ is the Sobolev space. An algorithm for finding analytically the unique solution $h\in\dot H^{-\alpha} (0,L)$ to (1) of minimal order of singularity is given. Here, $\dot H^{-\alpha}(0,L)$ is the dual space to $H^\alpha(0,L)$ with respect to the inner product of $L^2(0,L)$. Under suitable assumptions it is proved that $R:\dot H^{-\alpha}(0,L) \to H^\alpha(0,L)$ is an isomorphism. Equation (1) is the basic equation of random processes estimation theory. Some of the results are generalized to the case of multidimensional equation (1), in which case this is the basic equation of random fields estimation theory.

In this paper we consider initial boundary value problems for quasilinear parabolic differential inclusions governed by general operators of Leray-Lions type and continuously perturbed subdifferentials. Our main goal is to show the existence of extremal solutions within a sector formed by appropriately defined upper and lower solutions. One of the main difficulties that arises in the treatment of the parabolic problems considered here is due to the fact that the underlying solution space does not possess lattice structure. Furthermore, the variational techniques that can be used for the corresponding stationary problems are not applicable here. The main tools used in the proof of our result are abstract results on nonlinear evolution equations, regularization, comparison and truncation techniques as well as special test function techniques.