Homotopy regularization for a high-order parabolic equation

In this work we study the solvability of the Cauchy Problem for a quasilinear degenerate high-order parabolic equation \begin{equation*} \left\{ \begin{tabular}{lcl} $u_t=(-1)^{m-1}\nabla\cdot(f^n(|u|)\nabla\Delta^{m-1}u)$ & &in $\mathbb{R}^N\times\mathbb{R}_+$, $u(x,0)=u_0(x)$& & in $\mathbb{R}^N$, \end{tabular} \right. \end{equation*} with $m\in\mathbb{N},\ m>1$ and $n>0$ a fixed exponent. Moreover, $f$ is a continuous monotone increasing positive bounded function with $f(0)=0$ and the initial data $u_0(x)$ is bounded smooth and compactly supported. Thus, through an homotopy argument based on an analytic $\varepsilon$-regularization of the degenerate term $f^n(|u|)$ we are able to extract information about the solutions inherited from the polyharmonic equation when $n=0$

A stationary population model with an interior interface-type boundary

We propose a stationary system that might be regarded as a migration model of some population abandoning their original place of abode and becoming part of another population, once they reach the interface boundary. To do so, we show a model where each population follows a logistic equation in their own environment while assuming spatial heterogeneities. Moreover, both populations are coupled through the common boundary, which acts as a permeable membrane on which their flow moves in and out. The main goal we face in this work will be to describe the precise interplay between the stationary solutions with respect to the parameters involved in the problem, in particular the growth rate of the populations and the coupling parameter involved on the boundary where the interchange of flux is taking place.This paper has been partially supported by Ministry of Economy and Competitiveness of Spain under research project PID2019-106122GB-I00.Publicad

Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches

This work has been partially supported by the Ministry of Economy and Competitiveness of Spain under research project MTM2012-33258.Publicad

Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators

This work has been partially supported by the Ministry of Economy and Competitiveness of Spain under research project MTM2012-33258.Publicad

Branching analysis of a countable family of global similarity solutions of a fourth-order thin film equation

The main goal in this article is to justify that source-type and other global-in-time similarity solutions of the Cauchy problem for the fourthorder thin film equation can be obtained by a continuous deformation (a homotopy path). This is done by reducing to similarity solutions (given by eigenfunctions of a rescaled linear operator B) of the classic bi-harmonic equation This approach leads to a countable family of various global similarity patterns of the thin film equation, and describes their oscillatory sign-changing behaviour by using the known asymptotic properties of the fundamental solution of bi-harmonic equation. These include, as a key part, the problem of multiplicity of solutions, which is under particular scrutiny.This work was partially supported by the Ministry of Economy and Competitiveness of Spain under research project MTM2012-33258

Qualitative analysis of a cooperative reaction-diffusion system in a spatiotemporally degenerate environment

In this paper, we are concerned with the cooperative system in which ∂tu−Δu=μu+α(x,t)v−a(x,t)up and ∂tv−Δv=μv+β(x,t)u−b(x,t)vq in Ω×(0,∞); (∂νu,∂νv)=(0,0) on ∂Ω×(0,∞); and (u(x,0),v(x,0))=(u0(x),v0(x))>(0,0) in Ω, where p,q>1, Ω⊂RN(N≥2) is a bounded smooth domain, α,β>0 and a,b≥0 are smooth functions that are T-periodic in t, and μ is a varying parameter. The unknown functions u(x,t) and v(x,t) represent the densities of two cooperative species. We study the long-time behavior of (u,v) in the case that a and b vanish on some subdomains of Ω×[0,T]. Our results show that, compared to the nondegenerate case where a,b>0 on Ω×[0,T], such a spatiotemporal degeneracy can induce a fundamental change to the dynamics of the cooperative system.Pablo Álvarez-Caudevilla was partially supported by the Ministry of Economy and Competitiveness of Spain under research project MTM2012-33258. Yihong Du was partially supported by the Australian Research Council. Rui Peng was partially supported by NSF of China (11271167, 11171319), the Program for New Century Excellent Talents in University, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and Natural Science Fund for Distinguished Young Scholars of Jiangsu Province

On a class of linear cooperative systems with spatio-temporal degenerate potentials

This paper analyses a class of parabolic linear cooperative systems in a cylindrical domain with degenerate spatio-temporal potentials. In other words, potentials vanish in some non-empty connected subdomains which are disjoint and increase in size temporally. Then, the vanishing subdomains for the potentials are not cylindrical. Following a similar idea to the semiclassical analysis behaviour, but done here for parabolic problems, under these geometrical assumptions, the asymptotic behaviour of the system is ascertained when a parameter, in front of these potentials, goes to infinity. In particular, the strong convergence of the solutions of the system is obtained using energy methods and the theory associated with the Γ-convergence. Also, the exponential decay of the solutions to zero in the exterior of the subdomains where the potentials vanish is achieved.This paper has been partially supported by the Ministry of Science and Innovation of Spain under research project PID2019-106122GB-I00

Existence of solutions for a system of coupled nonlinear stationary bi-harmonic Schrodinger equations

We obtain existence and multiplicity results for the solutions of a class of coupled semilinear bi-harmonic Schrodinger equations. Actually, using the classical Mountain Pass Theorem and minimization techniques, we prove the existence of critical points of the associated functional constrained on the Nehari manifold. Furthermore, we show that using the so-called fibering method and the Lusternik-Schnirerman theory there exist infinitely many solutions, actually a countable family of critical points, for such a semilinear bi-harmonic Schrodinger system under study in this work.The authors like to thank the anonymous Referee by his valuable suggestions, helpful comments which further improved the content and presentation of the paper. The first and third authors have been partially supported by the Ministry of Economy and Competitiveness of Spain under research project MTM2012-33258. The second author has been partially supported by the Ministry of Economy and Competitiveness of Spain and FEDER funds under research project MTM2013-44123-P.Publicad

Existence of positive solutions for a Brezis-Nirenberg type problems involving an inverse operator

This article concerns the existence of positive solutions for the second order equation involving a nonlocal term −∆u = γ(−∆)−1u + |u| p−1u, under Dirichlet boundary conditions. We prove the existence of positive solutions depending on the positive real parameter γ > 0, and up to the critical value of the exponent p, i.e. when 1 < p ≤ 2 ∗ − 1, where 2∗ = 2N N−2 is the critical Sobolev exponent. For p = 2∗ − 1, this leads us to a Brezis-Nirenberg type problem, cf. [5], but, in our particular case, the linear term is a nonlocal term. The effect that this nonlocal term has on the equation changes the dimensions for which the classical technique based on the minimizers of the Sobolev constant, that ensures the existence of positive solution, going from dimensions N ≥ 4 in the classical Brezis-Nirenberg problem, to dimensions N ≥ 7 for this nonlocal problem.This research was partially supported by the Ministry of Economy and Competitiveness of Spain, and by the FEDER under research projects MTM2016-80618-P and PID2019-106122GB-I00. P. Alvarez-Caudevilla was also supported by the Ministry of Economy and Competitiveness of Spain under research project RYC-2014-15284