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Register a new userFast diffusion on noncompact manifolds: well-posedness theory and connections with semilinear elliptic equations
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We investigate the well-posedness of the fast diffusion equation (FDE) in a wide class of noncompact Riemannian manifolds. Existence and uniqueness of solutions for globally integrable initial data was established in [5]. However, in the Euclidean space, it is known from Herrero and Pierre [20] that the Cauchy problem associated with the FDE is well posed for initial data that are merely in $ L^1_{mathrm{loc}} $. We establish here that such data still give rise to global solutions on general Riemannian manifolds. If, in addition, the radial Ricci curvature satisfies a suitable pointwise bound from below (possibly diverging to $-infty$ at spatial infinity), we prove that also uniqueness holds, for the same type of data, in the class of strong solutions. Besides, under the further assumption that the initial datum is in $L^2_{mathrm{loc}}$ and nonnegative, a minimal solution is shown to exist, and we are able to establish uniqueness of purely (nonnegative) distributional solutions, which to our knowledge was not known before even in the Euclidean space. The required curvature bound is in fact sharp, since on model manifolds it turns out to be equivalent to stochastic completeness, and it was shown in [13] that uniqueness for the FDE fails even in the class of bounded solutions on manifolds that are not stochastically complete. Qualitatively this amounts to asking that the curvature diverges at most quadratically at infinity. A crucial ingredient of the uniqueness result is the proof of nonexistence of distributional subsolutions to certain semilinear elliptic equations with power nonlinearities, of independent interest.
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We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $Mtimes mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is studied. We consider also elliptic equations on $M$ with similar conditions at infinity.
Let $M$ be a smooth manifold with boundary $partial M$ and bounded geometry, $partial_D M subset partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator on $partial M smallsetminus partial_D M$. We prove the regularity and well-posedness of the mixed Robin boundary value problem $$Pu = f mbox{ in } M, u = 0 mbox{ on } partial_D M, partial^P_ u u + bu = 0 mbox{ on } partial M setminus partial_D M$$ under some natural assumptions. Our operators act on sections of a vector bundle $E to M$ with bounded geometry. Our well-posedness result is in the Sobolev spaces $H^s(M; E)$, $s geq 0$. The main novelty of our results is that they are formulated on a non-compact manifold. We include also some extensions of our main result in different directions. First, the finite width assumption is required for the Poincar{e} inequality on manifolds with bounded geometry, a result for which we give a new, more general proof. Second, we consider also the case when we have a decomposition of the vector bundle $E$ (instead of a decomposition of the boundary). Third, we also consider operators with non-smooth coefficients, but, in this case, we need to limit the range of $s$. Finally, we also consider the case of uniformly strongly elliptic operators. In this case, we introduce a emph{uniform Agmon condition} and show that it is equivalent to the Gaa rding inequality. This extends an important result of Agmon (1958).
In this paper, we investigate the problem of blow up and sharp upper bound estimates of the lifespan for the solutions to the semilinear wave equations, posed on asymptotically Euclidean manifolds. Here the metric is assumed to be exponential perturbation of the spherical symmetric, long range asymptotically Euclidean metric. One of the main ingredients in our proof is the construction of (unbounded) positive entire solutions for $Delta_{g}phi_lambda=lambda^{2}phi_lambda$, with certain estimates which are uniform for small parameter $lambdain (0,lambda_0)$. In addition, our argument works equally well for semilinear damped wave equations, when the coefficient of the dissipation term is integrable (without sign condition) and space-independent.
Under structural conditions which are almost optimal, we derive a quantitative version of boundary estimate then prove existence of solutions to Dirichlet problem for a class of fully nonlinear elliptic equations on Hermitian manifolds.
We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a conjecture by Gauduchon which extends the Calabi conjecture; this was one of the original motivations to this work. We were also motivated by the fact that there had been increasing interests in fully nonlinear pdes from complex geometry in recent years, and aimed to develop general methods to cover as wide a class of equations as possible.
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