We are interested in the limit, as $m\to\infty,$ of the solution $u_m$ of the porous-medium equation $u_t = \Delta u^m$ in a bounded domain $\Omega$ with Neumann boundary condition, $\frac{\partial u^m}{\partial n}= g$ on $\partial\Omega,$ and initial datum $u(0)=u_0\geq 0.$ It is well known by now that this kind of limit turns out to be singular. In the case $g\equiv 0,$ it was proved that there exists an initial boundary layer ${\underline u}_0,$ the so-called mesa, and $u_m(t)\to {\underline u}_0$ in $L^1(\Omega),$ for any $t>0,$ as $m\to\infty.$ In this work, we generalize this result to the case of arbitrary $g\in L^2(\partial \Omega),$ we prove that the initial boundary layer is still ${\underline u}_0$ and in general (even in the regular case) the limit function is not a solution of a Hele--Shaw problem. There exists a time interval $I$ where the limit of $u_m,$ as $m\to\infty,$ is the unique solution of a Hele--Shaw problem and elsewhere, $u_m$ conveges to the constant function $\frac{1}{\vert\Omega\vert}(\int_\Omega u_0+ t\int_{\partial\Omega}g).$

This paper deals with finding multiple sign-changing solutions of the following class of quasilinear problems: $$ \begin{cases} - (r^{\alpha} |u'(r)|^{\beta}u'(r))^{'} = \lambda~ r^{\gamma} f(u(r)),~~ 0 < r < R\\ u(R)= u'(0)=0 , \end{cases} $$ where $\alpha,$ $\beta$ and $\gamma$ are given real numbers, $\lambda > 0$ is a parameter, $f: {\bf R} \to {\bf R}$ is some continuous function and $0 < R < \infty$. A result on existence of infinitely many sign-changing solutions is obtained by considering a family of associated initial value problems which are solved through a shooting argument and a counting of zeroes.

Using differential inclusions and viability theory we define sliding modes for (feedback) controlled semilinear differential equations in Banach spaces. We compare this definition with the equivalent control method for infinite-dimensional systems proposed by V. Utkin and Yu. Orlov. We show that if the sliding manifold satisfies suitable regularity hypotheses and the semigroup is compact, the projected evolution found by means of the equivalent control and our sliding mode do coincide.

This paper studies the asymptotic behaviour near blow-up points of solutions of the system $$ u_t = \Delta u + u^{p_1} v^{q_1} $$ $$v_t = \Delta v + u^{p_2} v^{q_2} $$ with nonnegative, bounded initial data. We derive estimates on the blow-up rates, then we prove a Liouville-type theorem and finally, making use of these results, we obtain the description of possible blow-up patterns.

We prove estimates for solutions of the Cauchy problem for the inhomogeneous wave equation on $\mathbb R^{1+n}$ in a class of Banach spaces whose norms depend only on the size of the space--time Fourier transform. The estimates are local in time, and this allows one, essentially, to replace the symbol of the wave operator, which vanishes on the light cone in Fourier space, with an inhomogeneous symbol, which can be inverted. Our result improves earlier estimates of this type proved by Klainerman--Machedon [4, 5]. As a corollary, one obtains a rather general result concerning local well-posedness of nonlinear wave equations, which was used extensively in the recent article [8].

We consider nonlinear parabolic equations with gradient-dependent nonlinearities, of the form $u_t-\Delta u=F(u,\nabla u)$. These equations are studied on smoothly bounded domains of ${\mathbb R}^N$, $N\geq 1$, with arbitrary (continuous) Dirichlet boundary data. Under optimal assumptions of (superquadratic) growth of $F$ with respect to $\nabla u$, we show that gradient blow-up occurs for suitably large initial data; i.e., $\nabla u$ blows up in finite time while $u$ remains uniformly bounded. Various extensions and additional results are given. We also consider some equations where the nonlinearity is nonlocal with respect to $\nabla u$, and show that gradient blow-up usually does not occur in this case.