Let $D$ be a bounded smooth domain in $\mathbb{R}^N$. It is well known that large solutions of an equation such as $\Delta u= u^p, \;p>1$ in $D$ blow up at the boundary at a rate $\phi(\delta)$ which depends only on $p$. (Here $\delta(x)$ denotes the distance of $x$ to the boundary.) In this paper we consider a secondary effect in the asymptotic behaviour of solutions, namely, the behaviour of $u/\phi(\delta)-1$ as $\delta \to 0$. We derive estimates for this expression, which are valid for a large class of nonlinearities and extend a recent result of Lazer and McKenna ([7]).

In this paper we consider some systems of partial differential equations with parameters and boundary data of Dirichlet type. Some sufficient conditions under which solutions of these systems continuously depend on parameters and boundary data are given. The proofs of the main results of this work are based on some variational methods.

We are concerned with the multiplicity of positive and nodal solutions of $$ \begin{align} &-\Delta u +\mu u =Q(x) |u|^{p-2} u+h(x)\quad\text{in}\,\, \Omega\\ &\qquad\qquad u \in H_0^1(\Omega), \end{align} $$ where $N\geq 3$, $2<p<\frac{2N}{N-2}$ $\mu >0$, $Q\in C(\overline{\Omega})$, and $0\not\equiv h\in L^2(\Omega)$. We show that if the maximum of $Q$ is achieved at exactly $k$ different points of $\Omega$, then for large enough $\mu$ the above problem has at least $k+1$ positive solutions and $k$ no

Existence results to elliptic equations with critical exponents are obtained by using critical-point methods and an orthogonality argument in Hilbert space.

One considers hyperbolic--parabolic systems $ u_t+(f(u))_x=(B(u)u_x)_x$ in one space dimension. For large periodic initial data and for a wide class of such systems, we establish the global existence of ``weak'' solutions. These results can be applied to general systems provided they admit a compact invariant domain. We develop the case of a particular $2\times 2$ system, the Keyfitz--Kranzer system.

In this work we study the asymptotic behaviour of solutions to the Korteweg--deVries--Burgers equation in the case when the initial data has different asymptotic limits at $\pm\infty .$ The method used is the one developed by Kawashima and Matsumura to discuss the asymptotic behaviour of travelling-wave solutions to Burgers equation.

We consider the initial value problem for the inviscid shallow-water equations in the case where a "front" is present, i.e., a boundary where the fluid depth tends to zero. Since the wave speed in shallow water behaves like the square root of the depth, this results in a degenerate hyperbolic system "on the edge" of change of type. It is shown that smooth solutions exist for smooth initial data.

We present a new result on the regularity of the velocity potential in transonic gas dynamics of two space dimensions. We prove that the velocity potential is $C^{\infty}$ in a neighborhood of a sonic point if the sonic line is locally $C^{\infty}, $ and if the velocity potential is known to be locally $C^4. $ Our main tool is Hörmander's energy method for the Tricomi operator, the commutator estimate of Kato and Ponce, and a result of Berezin for a degenerate hyperbolic equation.

We consider nonlinear periodic radial wave equations of the form $$ u_{tt}-u_{rr}-\frac{N-1}r u_r+g(t,r,u)=0,\qquad t\in\mathbb{R},\quad 0<r<R. $$ The main purpose of the paper is to characterize the nonlinearities $g$ such that the equation has no nontrivial periodic solutions in $\mathbb{R}^N$ within a given natural class of functions $u$. The proofs are based on a new integral identity which we introduce for the solutions of the wave equation.

We provide a complete study of the large-time behavior of all solutions of the equation $ u''+|u|^{p-1}u=|u'|^{q-1}u' $, $ t\geq 0 $, for all values of the exponents $ p,\, q>1 $. We find the existence of two critical curves $ {q=p}$, and $ {q=2p/(p+1)}$, separating three regions of the plane $ {(p,q)}$ for which the global behaviors exhibit a sharp contrast.

(i) If $ {q\geq p}$, all nontrivial solutions blow up in finite time at the rate $ {u'(t)\approx (T-t)^{-1/(q-1)}}$, except for an exceptional (unbounded) global solution, unique up to a sign and a time-translation.

(ii) If $ {2p/(p+1)<q<p}$, all nontrivial solutions blow up in finite time. They do so at the same rate as above, except for an exceptional solution, unique up to a sign and a time-translation, which blowsup at the (faster) rate $ {u'(t)\approx (T-t)^{-p/(p-q)}}$.

(iii) If $ {1<q \leq 2p/(p+1)}$, all nontrivial solutions have oscillatory finite time blow-up, and the energy blows up at the rate $ {E(t)\approx (T-t)^{-2/(q-1)}}$.

As a generalization of the case $ {q\geq p}$, we prove that the second-order ODE $ {u''=F(u,u')}$, $ {t\geq 0}$ admits global solutions for all values of $ {u(0)}$, whenever $ {F}$, loosely speaking, does not grow faster with respect to $ {u}$ than with respect to $ {u'}$, with $ {F}$ having the sign of $ {u'}$ for $ {u'}$ large. Among the techniques used in this paper are the method of variable time-translations, introduced in a previous work of the author, phase-plane analysis, Liapunov functions, and suitable shooting methods.

We consider harmonic maps between compact Riemannian manifolds $M$, $N$ of dimension $m$, $n$ respectively. In case $m \ge 3$ we show that any set of harmonic maps with the uniformly bounded $m$-energy is compact in $C^{\infty}(M,N)$. As a corollary we obtain the gradient estimate of harmonic maps.

We find both necessary and sufficient conditions on the nonnegative increasing functions $f$ and $g$ so that positive solutions of $$ \begin{matrix} u_t=\Delta u+h(u,v)\qquad &v_t=\Delta v+r(u,v) \qquad &\text {in }\Omega\times (0,T)\\ \frac{\partial}{\partial\nu} u=f(v) \qquad &\frac{\partial}{\partial\nu} v=g(u)\qquad &\text {on } \partial\Omega\times (0,T) \end{matrix} $$ exist globally in time. We assume throughout that $h$ and $r$ are nonnegative, smooth and $\frac{h(u,v)}u$, $\frac{r(u,v)}v$ are globally bounded.

We give an easy extension of the Poincaré Index Formula from the disc to any surface with boundary. In particular we show that the sum of the indices of the vector field at the critical points depends only on the Euler characteristic of the surface and on the behaviour of its trajectories in the boundary. Our theorem improves previous results on the same formula.