arXiv subjects

Erik Sarrion-Pedralva

Publications and source records attributed to Erik Sarrion-Pedralva.

3 recordsLinked to original sources

arXiv2026

Global self-similar solutions for Hardy-Hénon equations with linear and quasilinear diffusion

Global self-similar solutions to the parabolic Hardy-Hénon equation $$ u_t=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ are classified in the range of exponents $m\geq1$, $p>m$ and $σ>\max\{-2,-N\}$. The classification varies strongly with respect to the celebrated \emph{Fujita} and \emph{Sobolev critical exponents} $$ p_F(σ)=m+\frac{σ+2}{N}, \quad p_S(σ)= \begin{cases} \frac{m(N+2σ+2)}{N-2}, & \mbox{if } N\geq3, \\[1mm] \infty, & \mbox{if } N\in\{1,2\}. \end{cases} $$ Indeed, if $p\in(p_F(σ),p_S(σ))$, both equations admit self-similar solutions with either compact support (if $m>1$) or Gaussian-like tail as $|x|\to\infty$ (if $m=1$), as well as a one-parameter family satisfying $$ u(x,t)\sim C|x|^{-(σ+2)/(p-m)}, \quad {\rm as} \ |x|\to\infty. $$ If $p\geq p_S(σ)$, there are only self-similar solutions with the latter algebraic tail, while for $m<p\leq p_F(σ)$ no global solutions exist. The results open the way for a deeper study of the role of these solutions in the dynamics of the Hardy-Hénon equations.

math.AP↗

arXiv2021

First Dirichlet eigenvalue and exit time moment spectra comparisons

We prove explicit upper and lower bounds for the Poisson hierarchy, the averaged $L^1$-moment spectra $\{\dfrac{\mathcal{A}_k\left(B_R^M\right)}{\text{vol}\left(S_R^M\right)}\}_{k=1}^\infty$, and the torsional rigidity $\mathcal{A}_1(B^M_R)$ of a geodesic ball $B^M_R$ in a Riemannian manifold $M^n$ which satisfies that the mean curvatures of the geodesic spheres $S^M_r$ included in it, (up to the boundary $S^M_R$), are controlled by the radial mean curvature of the geodesic spheres $S^ω_r(o_ω)$ with same radius centered at the center $o_ω$ of a rotationally symmetric model space $M^n_ω$. As a consecuence, we prove a first Dirichlet eigenvalue $λ_1(B^M_R)$ comparison theorem and show that equality with the bound $λ_1(B^ω_R(o_ω))$, (where $B^ω_r(o_ω)$ is the geodesic $r$-ball in $M^n_ω$), characterizes the $L^1$-moment spectrum $\{\mathcal{A}_k(B^M_R)\}_{k=1}^\infty$ as the sequence $\{\mathcal{A}_k(B^ω_R)\}_{k=1}^\infty$ and vice-versa.

math.DG↗

arXiv2021

First Eigenvalue of the Laplacian of a Geodesic Ball and Area-Based Symmetrization of its Metric Tensor

Given a Riemmanian manifold, we provide a new method to compute a sharp upper bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of radius less than the injectivity radius of the manifold. This upper bound is obtained by transforming the metric tensor into a rotationally symmetric metric tensor that preserves the area of the geodesic spheres. The provided upper bound can be computed using only the area function of the geodesic spheres contained in the geodesic ball and it is sharp in the sense that the first eigenvalue of geodesic ball coincides with our upper bound if and only if the mean curvature pointed inward of each geodesic sphere is a radial function.

math.DG↗