The course aims at presenting an introduction to the subject of singularity formation in nonlinear evolution problems usually known as blowup. In short, we are interested situation where, starting from a smooth initial configuration, and after first period classical evolution, solution (or some cases its derivatives) becomes infinite finite time due cumulative effect nonlinearities. We concentrate on involving differential equations parabolic type, or systems such equations.<br> A part...

Abstract We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion: \input amssym $$\left\{ {\matrix{ {{{\partial u} \over {\partial t}} + \left( { ‐ \Delta } \right)^{\sigma /2} {\left| u \right|^{m 1} \right) = 0,} \hfill &amp; {x \in {\Bbb R} ^N ,\,\,t &gt; \cr {u\left( {x,0} f\left( x \right),} .} \right.$$ consider data $f\in L^1(\Bbb{R}^N)$ all exponents $0&lt;\sigma&lt;2\;and\;m&gt;0$ . Existence strong solution is established $...

The possible continuation of solutions the nonlinear heat equation in RN × R+ ut = Δum + up with m > 0, p 1, after blowup time is studied and different modes are discussed terms exponents p. Thus, for ≤ 2 we find a phenomenon nontrivial where region {x : u(x, t) ∞} bounded propagates finite speed. This call incomplete blowup. For N ≥ 3 m(N 2)/(N − 2) that blow at t T then become again T. Otherwise, complete wide class initial data. In analysis behavior large p, list critical appears whose...

We establish the uniqueness of fundamental solutions to p -Laplacian equation \mathrm {(PLE)} \; u_t = {div} (|Du|^{p-2}Du), &gt; 2, defined for x \in \mathbb R^N , 0 &lt; t T . derive from this result asymptotic behaviour nonnegative with finite mass, i.e. such that u(\cdotp, t) L^1(\mathbb R^N) Our methods also apply porous medium {(PME)} \Delta (u^m), m 1, giving new and simpler proofs known results. finally introduce yet another method proving results based on idea radial symmetry. This...

We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. establish existence uniqueness results in a suitable class of solutions using the theory maximal monotone operators dual spaces. Then we describe long-time asymptotics terms separate-variables friendly giant type. As by-product, obtain an result for semilinear elliptic non local with sub-linear The Appendix...

We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range effects. Our main concern is so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some its generalizations. Contrary to usual flows, version has infinite speed propagation for all exponents $0 0$; other hand, it also generates an $L^1$-contraction semigroup which depends continuously exponent differentiation nonlinearity. After establishing general...

We establish the existence, uniqueness and main properties of fundamental solutions for fractional porous medium equation introduced in [51]. They are self-similar functions form u(x,t) = t^{–\alpha} f(|x| t^{–\beta}) with suitable α β . As a application this construction, we prove that asymptotic behaviour general is represented by such special solutions. Very singular also constructed. Among other interesting qualitative an Aleksandrov reflection principle.

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is u_t=\nabla\cdot(u\nabla (-\Delta)^{-s}u), \quad \ 0&lt;s&lt;1. problem posed in \{x\in\mathbb R^n, t\in \mathbb R\} nonnegative initial data u(x,0) that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, weak solutions satisfying energy estimates finite propagation. As main results we establish...