We present a one-dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions we show that gradient blow-up occurs at a single point, and we study the shape of the singularity. Our argument for gradient blow-up also provides a pair of "naive viscosity sub- and super-solutions" which violate the comparison principle.

We study a system of nonlinear dispersive equations of the form $$ \partial_t u_k + \partial_x^{2j+1}u_k + F_k(u_1,\ldots ,u_n,\ldots , \partial_x^{2j}u_1, \ldots , \partial_x^{2j} u_n)=0, (t,x) \in \mathbf{R}\times h{\mathbf{R}}, $$ where $k=1,\ldots ,n$, $j\in \mathbf{N}$ and $F_k(\cdot)$ is a polynomial having no constant or linear terms. Local existence of solutions to the associated initial value problem is shown without a smallness condition on the data.

This paper establishes the blowup estimate near the blowup time for the heat equation $u_t = \Delta u$ with the nonlinear boundary condition $u_n = u^p$ on $\partial \Omega\times [0,T)$, where $\Omega$ is a bounded domain in $R^N$. It is proved that the blowup will not occur in the interior of the domain. The blowup rate is established under certain conditio

We take into consideration several minimization problems and we study the asymptotic behavior of minimizers. We introduce corresponding notions of renormalized energies and we give explicit formulas in the case of a ball. We study the Ginzburg-Landau energy in an appropriate class of functions and we find the link with the renormalized energies already introduced.

This article is concerned with travelling-wave solutions of the nonlinear reaction-\newline convection-diffusion equation $$ u_t=(a(u))_{xx}+(b(u))_x+c(u) $$ in which subscripts denote partial differentiation. The coefficients $a$, $b$ and $c$ are supposed to be defined in some closed real interval with infimum 0 and supremum $\ell,0<\ell\leq \infty$. The search is for nontrivial travelling-wave solutions of the form $u(x,t)=f(x-\lambda t)$ where $\lambda$ denotes the wave speed. Specifically, solutions of this type, where $f$ is defined and nonincreasing on $(0,\infty)$ with $f(\xi)\rightarrow 0$ as $\xi \rightarrow \infty$, and where $f$ is defined and nonincreasing on $(-\infty,\infty)$ with $f(\xi)\rightarrow 0$ as $\xi\rightarrow\infty$ and $f(\xi)\rightarrow\ell$ as $\xi\rightarrow-\infty$, are sought. Under mild continuity assumptions on the coefficients $a$, $b$ and $c$ it is shown that the existence of such a travelling wave is equivalent to the existence of a solution of a singular nonlinear integral equation. A number of applications of this result are discussed. In particular the correspondence between the travelling waves and the integral equation is used to resolve an open question in the theory of finite speed of propagation for the reaction-convection-diffusion equation

For the problem $\Delta u + f(u)=0 \ \text{ in } \ \Bbb R^n$; $u(x)\rightarrow 0, \ \text{as} \ |x| \rightarrow \infty$ we use a shooting method to prove that there is at most one positive radially symmetric solution if $u$ decays like $|x|^{-(n-2)}$ as $|x| \rightarrow \infty$, and $f$ is similar in shape to $f(u)=u^p-u^q$ with $n>2$ and $q>p>(n+2)/(n-2)$.

The goal of this paper is to study the existence of solutions for the following class of problems for the $N$-Laplacian $$ u\in W_0^{1,N}(\Omega),\quad u\geq 0\ \ \text{ and } \ \ -\Delta_Nu\equiv -\text{div}( | \nabla u | ^{N-2}\nabla u)=f(x,u) \,\,\, \rm{in}\,\,\, \Omega , $$ where $\Omega $ is a bounded smooth domain in $\Bbb R^N$ with $N\geq 2$ and the nonlinearity $f(x,u)$ behaves like $\exp (\alpha | u | ^{\frac N{N-1}})$ when $ | u | \to \infty .$

We study the quasilinear elliptic equation $$ \left\{\eqalign{&\text{div}|\nabla u|^{m-2}+K(|x|)|u|^{q-1}=0,\quad x\in\mathbb{R}^n,\cr&\lim_{|x|\to \infty}|x|^{{n-m\over m-1}} | u(|x|) | >0,\cr}\right. $$ where $1<m<n,$ $ q>m-1,$ $ K(r)\in C^{1}((0,\infty))$ and $ K(r)>0 $ on $ (0,\infty).$ We show the existence of radial solutions with prescribed numbers of zeros under simple conditions. These results are generalizations of those due to E\. Yanagida and S\. Yotsutani ([17]) and Y. Naito ([13]) for $m=2$, but the proofs are considerably different from theirs even if $m=2$. Finally, we consider various further boundary problems with similar results.

We study the asymptotic behavior of solutions to the---generally nonautonomous and nonlinear---Cauchy problem $$ u^{\prime}(t) \in A(t)u(t) + f(t), \ \ t \in \mathbb{R}^+ , \,\,\, u(0) = u_0. \tag {$CP_t$} $$ The emphasis is on almost-periodicity properties of the solution, in particular on weak almost periodicity (in the sense of Eberlein).

We obtain, under weaker assumptions, a result of Dancer, Lazer-McKenna on the existence of at least three solutions for some elliptic problems with jumping nonlinearities by means of simple variational tools.

We use degree results of Dancer and Du ([10]) and bifurcation and monotonicity techniques of Brown and Du ([11]) to study a competitor-competitor-mutualist model with diffusion. We obtain rather complete results on the existence and nonexistence of positive periodic solutions. It turns out that this model behaves largely as the two-species competition model.

Two problems for the abstract quasilinear evolution equation of "hyperbolic" type in a Banach space $$ u'(t) = A(t,u(t))u(t) \,\,\,\rm{for} \,\,\,t \geq 0, \,\,\, \rm{and} \,\,\, u(0)=u_0 $$ are studied without assuming that the domain of $A(t,w)$ is dense in the whole space. One is the fundamental problem of existence and uniqueness of classical solutions, and the other is the problem of extension or blow up of classical solutions.

In this paper we consider the equation (E) $ \partial _{t} \Psi = \Delta \Psi + a \mid \nabla \Psi \mid ^{q} + \mid \Psi \mid^{p-1}\Psi ,$ $ t >0 , $ $ x\in \Bbb R^{n},$ which was first studied by Chipot and Weissler in a bounded domain with $a = -1 $. For $a$ sufficiently small, $q = \frac{2p }{ p+1 }$ the critical exponent and $1<{ n(p-1) \over 2 } < p+1,$ we establish the existence of a positive global self-similar solution of (E) with a singular initial data at the origin. In particular, this implies a nonuniqueness result for the Cauchy problem associated with (E) in $L^{s}( \Bbb R ^{n})$, where $ 1 \leq s < {n(p-1) \over {2}},$ and $p,$ $q$ and $a$ are as above. Also, for $a<0$ and $1<{ n(p-1) \over 2 } < p,$ an explicit estimate is given for the range of allowed values of $a.$

We prove estimates, in weighted $L^\infty$ spaces, for solutions of the BGK equation in the whole space and lower bound on the associated macroscopic density. $L^\infty$ bound on the macroscopic object $\rho, u$ and $T$ are deduced. Then we may show uniqueness of the solution of the BGK equation with $L^\infty$-bound assumption on the initial data, propagation of estimates on derivatives. As an application, with a BV-bound assumption on the initial data we get the convergence with rate $(\Delta t)^{1/2}$ of a time semi-discretized scheme to the solution.

We prove the existence of a global attractor for the equation $$ u_{tt} + u_t - \Delta u + f(u) = 0 ,\quad u= u(x,t) \ , \ x\in \mathbb{R}^N . $$ The attractor is compact in $ H^1_{loc}(\mathbb{R}^N .) \times L^2_{loc}(\mathbb{R}^N . ).$

We prove that all the positive harmonic functions on the upper half space $ \{ x : x= (x_{1}, \cdots, x_{n} ), x_{n} \geq 0 \} $ $ (n \geq 3) $ satisfying the boundary condition $ D_{x_n} (u) = - u^{n/(n-2)} $ are fundamental solutions of the Laplace equation multiplied by proper constants. We also prove that there is no positive harmonic function on the upper half space satisfying the subcritical boundary condition $ D_{x_n} (u) = - u^{p} $ for $p<n/(n-2).$

We consider the singularly perturbed problem (*) $$\begin{cases} \epsilon\dot u & = Ju+\epsilon^p g(u,v,\epsilon), \\ \dot v & = f(u,v,\epsilon) \end{cases} $, $J= \begin{pmatrix} 0 &-1 \\ 1 &0 \end{pmatrix}$$, assuming that the degenerate system $\dot v=f(0,v,0)$ has an orbit $v_0(t)$ homoclinic to the hyperbolic equilibrium $v=0$. Under certain conditions on $f, g$ (that are stated in Section 2), we show that the bifurcation towards a homoclinic orbit $v(t,\epsilon)$ of (*) depends on three bifurcation functions $G_i(\alpha ,\epsilon), \, i=1,2,3, \, \alpha \in \mathbb{C} $, that are $2\pi \epsilon$-periodic in $\alpha $, for $\epsilon\neq 0$, and satisfy $|G_i(\alpha ,\epsilon)-G_i^0(\epsilon)| \leq C\epsilon^{-1} e^{-\eta _0/|\epsilon|}$, $\eta _0 >0$, where $G_i^0(\epsilon)={1\over {2\pi \epsilon}}\int_{0}^{2\pi \epsilon} G_i(\alpha ,\epsilon)\, d\alpha$. Thus we see that if $G_i^0(\epsilon)=0$ the bifurcation functions are exponentially small. This fact is then used to recover some of the results shown in [4].