We show that general systems of elliptic differential operators have a bounded $H_\infty$-functional calculus in $L_p$ spaces, provided the coefficients satisfy only minimal regularity assumptions.

We investigate explosive solutions of the equation $\Delta u=f(u)$ in a bounded convex domain $D$. We find some asymptotic estimates and prove the log-convexity of the solution $u(x)$ under suitable conditions on $f$.

The oscillator $$ x''+h(t)x'+x=0 $$ is considered, where the damping $h:{\Bbb R}_+\to{\Bbb R}_+$ is piecewise continuous and large in the sense $$ \liminf_{t\to\infty}\int _t^{t+\delta}h>0 \quad \text{ for every }\ \delta>0. $$ The problem of intermittent damping, initiated by P\. Pucci and J\. Serrin, is investigated. Let a sequence $\{I_n=[\alpha_n,\beta_n]\}$ of disjoint intervals be given such that $\alpha_n\to\infty$ as $n\to\infty$. A necessary and sufficient condition is given for $\{I_n\}$ and $h$ on $I:=\bigcup _{n=1}^\infty I_n$ guaranteeing $x(t)\to 0$, $x'(t)\to 0$ as $t\to\infty$ for every solution $x$, any way $h$ may be defined out of $I$.

In this paper we study the existence, uniqueness and asymptotic behaviour of the solution for a class of first-order symmetric hyperbolic systems introduced by Friedrichs [(6)], in a bounded domain with an energy-absorbing boundary. It is carried out using the Fourier transform method. An energy decay rate result is obtained.

Existence and nonexistence results of small solutions are derived for certain boundary value problems.

We consider abstract nonlinear second-order-in-time systems with damping. The nonlinearity is assumed to satisfy a monotonicity condition as well as certain smoothness conditions. Well posedness of solutions is established and several examples of interest are discussed. A nonlinear variation-of-parameters representation for solutions in terms of an associated linear semigroup is also given.

We determine necessary and sufficient conditions for a planar system of the form $$ \begin{align} \frac{dx}{dt} & = ax + by + x ( \alpha x + \beta y + Ax^{2} + Bxy + Cy^{2}) \\ \frac{dy}{dt} & = cx + dy + y ( \alpha x + \beta y + Ax^{2} + Bxy + Cy^{2}) \, , \end{align} $$ where $a, b, c, d, \alpha , \beta , A, B$ and $C$ are real constants, to possess a centre at the origin. The approach for computing the necessity of the conditions (i.e., for the calculation of the focal values) is somewhat unconventional, since it relies on the properties of an integrating factor, rather than a first integral, which expedites the procedure. The subclass consisting of those systems which possess a centre at the origin is studied further. In particular, first integrals are given, and phase portraits are drawn of all topologically inequivalent nonlinear members.

We extend some earlier results on existence of homoclinic solutions for a class of Hamiltonian systems. We also study heteroclinic solutions. We use a variational approach.

The diffusion equation with a concentrated singular reaction $v_t=v_{xx} + \epsilon\delta(x-a)f(v)$ $(\epsilon>0, 0<a<1)$ is studied. Criteria for global existence and finite time quenching of the solution are established. The growth rate and estimate on quenching time are also given for a certain class of nonlinearities.

In this paper we look for homoclinic solutions of the system $$ \ddot q-a(t)\mid q\mid ^{p-2}q+W_q(t,q)=0 $$ where $p>2,$ $a(t)\rightarrow +\infty $ as $\mid q\mid \rightarrow +\infty $ and $W(t,\cdot )$ is even and quadratic or superquadratic at infinity and at the origin. Using a compact embedding between suitable weighted Sobolev spaces, we prove the existence of infinitely many homoclinic solutions of the problem.

We prove Sobolev interpolation inequalities for the Stokes operator, giving sharp $L^{\infty}$ estimates for solenoidal vector fields in $\mathbb{R}^2$ and $\mathbb{R}^3$. We use fundamental solutions of a generalized Stokes system in the proofs.