Moser-type estimates for solutions of linear elliptic equations with lower-order terms are given. The results are proved under various assumptions on the coefficients and on the known term in the equations. Also solutions of Hamilton-Jacobi equations are considered.

We consider the Cauchy problem for the fast diffusion equation: $$ \partial_t u = \text{{\rm div}}(|\nabla u|^{p-2}\nabla u) \text{ in } \mathbb{R}^N \times (0, \infty ) \text{ and } u(x,0) =\varphi (x) \text{ in } \mathbb{R}^N $$ with $ 1 <p < 2$ for nonzero bounded nonnegative initial data $\varphi$ having compact support, and show that the set of positive spatial critical points of the nonnegative solution is contained in the closed convex hull of the support of the initial datum for any time either when $\frac 32 < p <2$ and $N \geqq 2$ or when $1 < p <2$ and $N = 1$, and further in the case $ N = 1 $ it consists of one point after a finite time.

The purpose of this paper is to study the critical value of the quenching problem for a semilinear heat equation with negative power reaction term which depends also on the spatial variable. This problem arises when we deal with nonhomogeneous media. We show that the critical value exists and is finite. Moreover, we also give some estimates of the critical value.

We consider a beam equation with a nonlocal nonlinearity of Kirchhoff type on an unbounded domain. We show that smooth global solutions decay (in time) at a uniform rate as $t\to +\infty$. Our model is closely related to a nonlinear Schrödinger equation with a time-dependent dissipation. We use this observation to obtain intermediate information on our original model.

Using the lower and upper solutions method together with degree theory, we study the existence and multiplicity of positive solutions for the problem $$ (\varphi_{p}(u'))'+f(t,u)=0,\ \ a_{1}\varphi_{p}(u(a))-a_{2} \varphi_{p}(u'(a))=0,\ b_{1}\varphi_{p}(u(b))+b_{2} \varphi_{p}(u'(b))=0, $$ where $\varphi_{p} (s):=|s|^{p-2}s, \,p>1$, $a_1,b_1\in\Bbb R$, $a_2,b_2\in\Bbb R^+$, $a_1^2+a_2^2>0$, $b_1^2+b_2^2>0.$ The function $f$ satisfies assumptions related to the classically called sublinear, superlinear, subsuperlinear, or supersublinear cases. Our results improve the recent ones of L.H. Erbe-H. Wang ([21]) and L.H. Erbe-S. Hu-H. Wang ([20]).

Let us consider the resonant boundary value problem $$ \begin{align} - &u''(x) - u(x) + g(u(x)) = h(x), \quad x \in [0,\pi], \\ &u(0) = u(\pi) = 0, \end{align} $$ where $ g: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous and $ T$-periodic function with zero mean value, not identically zero, and $ h \in C[0,\pi].$ If each $ h \in C[0,\pi] $ is written as $ h(x) = a\sin x + \tilde{h}(x), $ where $ a \in \mathbb{R} $ and $\int_{0}^{\pi} \tilde{h}(x) \sin x \ dx = 0,$ then, it is shown that for each $ \tilde{h},$ there are real numbers $ a_{1}(\tilde{h}) < 0 < a_{2}(\tilde{h})$ (which depend continuously on $ \tilde{h}$), such that there is solution if and only if $ a\in [a_{1}(\tilde{h}),a_{2}(\tilde{h})].$ In relation to the multiplicity, it is proved that the number of solutions increases to infinity as $ a $ goes to zero. The proof combines different tools such as Liapunov-Schmidt reduction and upper-lower solutions notions, together with a careful analysis of the connected subsets of the solution set of the auxiliary equation in the alternative method, as well as a detailed study of the oscillatory behavior of some integrals associated to the bifurcation equation of the previous problem.

In view of the lack of a global regularity theorem for the solutions $(v,p)$ of the Navier-Stokes equations there has been a great deal of activity in establishing sufficient conditions on the velocity $v$ in order to guarantee the regularity of the solution. However, nontrivial conditions involving the pressure seem not to be available in the literature. In this paper we present a sharp sufficient condition involving a combination of $v$ and $p$. The proof relies on the truncation method, introduced in reference [3] for studying scalar elliptic equations and developed further by many authors (see, in particular [6] and [4]). In the sequel we use some basic results proved in [4].

In this paper we study the problem $-\Delta u = u^{p} + \lambda u^{q}, u > 0$ in $B(1), u = 0$ on $\partial B (1)$ and $\lambda > 0.$ We show that in the case $1 < p \leq {(n+2)/(n-2)}$ there are at most two solutions if $q = 0$, while there are exactly two solutions if $\lambda$ is sufficiently small and $0 < q < 1.$ In the supercritical case $\bigl( p > {(n+2)/(n-2)}\bigr)$ we show the nonexistence of solutions for $1 \leq q < {(n+2)/(n-2)}$ and $\lambda$ sufficiently small. We also consider the problem $- \Delta u = \lambda [ (1 + \alpha u)^{p} + \mu (1 +\alpha u)], u > 0$ in $B(1), u = 0$ on $\partial B(1), \lambda > 0, \alpha > 0$ and $\mu \geq 0.$ We show that for $1 < p \leq {(n+2)/(n-2)}$, there are at most two solutions and, actually, exactly two for $\lambda$ small.

We study a 2-phase free boundary problem, in which the positive and negative parts of a solution satisfy two different elliptic equations, and a condition, involving normal derivatives from positive and negative sides holds on the free boundary in a weak sense. We show that if the free boundary is locally a graph of Lipschitz function, then it is $C^{1,\alpha} $ smooth. This is an extension of the result obtained by L.Caffarelli in the case when the positive and negative parts of a solution satisfy the same elliptic equation.

We prove nonlinear stability of planar shock front solutions for certain viscous scalar conservation laws in two space dimensions. Let us admit the planar shock front solution in x-direction, where the flux function is only in the x-direction and its convexity is not necessarily assumed. For either the nondegenerate or degenerate cases, if the initial disturbance is sufficiently small, then the solution approaches to the shifted planar shock front solution as t$ \rightarrow \infty$. Here, the shift function may have different asymptotic states in $y$-direction. The proofs are given by applying an elementary weighted energy method to the ``integrated equation" which is equivalent to the original one.

A classical solution of the damped Boussinesq equation $$ u_{tt}-2bu_{txx}=-\alpha u_{xxxx}+u_{xx}+\beta (u^2)_{xx},\quad x\in {\Bbb R}^1,t>0, $$ with $\alpha ,b=\text{const}>0$, $\beta =\text{const}\in{\Bbb R}^1$, $\alpha >b^2$, and small initial data is constructed by means of the successive application of the spectral theory and the perturbation one. Its long-time asymptotic representation is obtained which shows that the major term increases linearly with time and the second term contains a combination of the Airy functions of a negative argument. A uniform-in-space estimate of the remainder is given.