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Abstract We consider the porous medium equation with power-type reaction terms u p on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If p > m , small data give rise to global-in-time solutions while solutions associated to large data blow up in finite time. If p < m , large data blow up at worst in infinite time, and under the stronger restriction p ∈ (1 , (1 + m) / 2 ] all data give rise to solutions existing globally in time, whereas solutions corresponding to large data blow up in infinite time. The results are in several aspects significantly different from the Euclidean ones, as has to be expected since negative curvature is known to give rise to faster diffusion properties of the porous medium equation. [ABSTRACT FROM AUTHOR]

We consider nonnegative solutions of the porous medium equation (PME) on Cartan–Hadamard manifolds whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we estimate the location of the free boundary. A global Harnack principle follows. We also present a change of variables that allows to transform radially symmetric solutions of the PME on model manifolds into radially symmetric solutions of a corresponding weighted PME on Euclidean space and back. This equivalence turns out to be an important tool of the theory. [ABSTRACT FROM AUTHOR]

We consider a Schrödinger hamiltonian H ( A , a ) with scaling critical and time independent external electromagnetic potential, and assume that the angular operator L associated to H is positive definite. We prove the following: if ‖ e − i t H ( A , a ) ‖ L 1 → L ∞ ≲ t − n / 2 , then ‖ | x | − g ( n ) e − i t H ( A , a ) | x | − g ( n ) ‖ L 1 → L ∞ ≲ t − n / 2 − g ( n ) , g ( n ) being a positive number, explicitly depending on the ground level of L and the space dimension n . We prove similar results also for the heat semi-group generated by H ( A , a ) . [ABSTRACT FROM AUTHOR]

We consider two-dimensional Schrödinger operators H with an Aharonov–Bohm magnetic field and an additional electric potential. We obtain an explicit leading term of the asymptotic expansion of the unitary group for in weighted -spaces. In particular, we show that the magnetic field improves the decay of with respect to the unitary group of non-magnetic Schrödinger operators, and that the decay rate in time is determined by the magnetic flux. [ABSTRACT FROM AUTHOR]

Abstract: We study a class of nonlinear diffusion equations whose model is the classical porous media equation on domains , , with homogeneous Neumann boundary conditions. We improve the known results in such model case, proving sharp – regularizing properties of the evolution for short time and sharp long time bounds for convergence of solutions to their mean value. The generality of the discussion allows to consider, almost at the same time, weighted versions of the above equation provided an appropriate weighted Sobolev inequality holds. In fact, we show that the validity of such weighted Sobolev inequality is equivalent to the validity of a suitable – bound for solutions to the associated weighted porous media equation. The long time asymptotic analysis relies on the assumed weighted Sobolev inequality only, and allows to prove uniform convergence to the mean value, with the rate predicted by linearization, in such generality. [Copyright &y& Elsevier]

Abstract: Consider a p-homogeneous functional () and suppose that a weighted Poincaré inequality involving it holds. Then all solutions to the evolution equation driven by the associated weighted p-Laplacian satisfy, for any , , the bound . Such bound is in fact equivalent to the Poincaré inequality. There are examples in which the Poincaré inequality holds but the evolution does not map into for any t and any . Moreover, if a p-logarithmic Sobolev inequality holds then the Poincaré inequality is shown to hold too, therefore the previous regularization result is valid. Finally, the weighted Sobolev-type inequality () implies – regularization of the evolution for any , all and an explicit . There are cases in which the weighted Sobolev-type inequality holds but the associated evolution does not map into for any , , . [Copyright &y& Elsevier]

Abstract: Let M be a compact Riemannian manifold without boundary. Consider the porous media equation , , being the Laplace–Beltrami operator. Then, if , the associated evolution is regularizing at any time and the bound holds for for suitable explicit . For large t it is shown that, for general initial data, approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but approaches with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting. [Copyright &y& Elsevier]

Let X be a locally compact metrizable space endowed with a couple of equivalent finite Radon measures m and μ and let E be a Hilbert C&ast;-monomodule over C0(X). We consider a class of abstract nonlinear parabolic equations defined as follows. Let ∂ be a closed derivation from L2(X,m) to L2(E,μ) and Tt be the strongly continuous nonlinear semigroup naturally associated, in the sense of Brezis (1973), to the convex l.s.c. functional E(u)=∫X|∂u|xp dμ(x), where |·| is the natural modulus function associated to E. The generator of the semigroup considered is a natural generalization of the usual p-Laplacian operator. We suppose that a suitable Sobolev-like inequality of the form &par;u&par;L2d/(d−2)(X,m)&les;c&par;∂u&par;L2(E,μ) holds true for some d>2, with p∈[2,d). Then Tt is a nonlinear Markov semigroup in the sense that it is order preserving and nonexpansive on each Lq(X,m) for any q∈[2,+∞] and, moreover, it satisfies &par;Ttu−Ttv&par;L∞(X,m)&les;cm(X)αt−β&par;u−v&par;γLq(X,m) for all q&ges;2 and suitable constants α, β, γ depending only on p, q, d. Examples include the semigroup generated by the p-Laplacian on finite measure manifolds with boundary and homogeneous Dirichlet boundary conditions, as well as by p-Laplacian-like operators associated both to regular sub-Riemannian structures, and to systems of (possibly singular or degenerate) vector fields satisfying the appropriate Sobolev inequalities. [Copyright &y& Elsevier]

We prove existence and uniqueness of distributional, bounded solutions to a fractional filtration equation in R d. With regards to uniqueness, it was shown even for more general equations in [22] that if two bounded solutions u , w of (1.1) satisfy u − w ∈ L 1 (R d × (0 , T)) , then u = w. We obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions. For nonnegative initial data, we first show that a minimal solution exists and then that any other solution must coincide with it. A similar procedure is carried out for sign-changing solutions. As a consequence, distributional solutions have locally-finite energy. [ABSTRACT FROM AUTHOR]

We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian − Δ H N − ( N − 1 ) 2 / 4 on the hyperbolic space H N , ( N − 1 ) 2 / 4 being, as it is well-known, the bottom of the L 2 -spectrum of − Δ H N . We find the optimal constant in a resulting Poincaré–Hardy inequality, which includes a further remainder term which makes it sharp also locally: the resulting operator is in fact critical in the sense of [17] . A related improved Hardy inequality on more general manifolds, under suitable curvature assumption and allowing for the curvature to be possibly unbounded below, is also shown. It involves an explicit, curvature dependent and typically unbounded potential, and is again optimal in a suitable sense. Furthermore, with a different approach, we prove Rellich-type inequalities associated with the shifted Laplacian, which are again sharp in suitable senses. [ABSTRACT FROM AUTHOR]