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Alexandre Arias Junior

Publications and source records attributed to Alexandre Arias Junior.

13 recordsLinked to original sources
arXiv2026

Degenerate 3-evolution equations in Gevrey classes

We consider the Cauchy problem for third-order evolution differential operators with variable coefficients, depending on time $t\in [0,T]$ and space $x\in\mathbb{R}$, where the leading coefficient $a_3(t)$ vanishes at $t = 0$ with finite order. We establish sufficient conditions on the behavior of the lower order coefficients $a_j(t,x)$ $j=1,2$ as $t \to 0^{+}$ and $|x| \to \infty$ that ensure well-posedness in $L^2(\mathbb{R})$, $H^{\infty}(\mathbb{R})$ and Gevrey-type spaces.

math.AP↗
arXiv2025

Smoothing effect for higher order dispersive equations and applications to nonlinear initial value problems

In this paper we deal with the initial value problem related to a family of dispersive inhomogeneous evolution equations Pu=f with variable coefficients belonging to the class of p-evolution equations, $p\geq 2$. We study the smoothing effect produced by some spatial decay assumptions on the imaginary part of the subleading coefficient of the linear operator P. Then we apply this result to nonlinear problems with derivative nonlinearities obtaining existence and uniqueness of the solution in a suitable Sobolev class. The nonlinear equations considered include various equations of physical interest such as KdV-type and Kawahara-type equations.

math.AP↗
arXiv2024

Polynomially oscillatory multipliers on Gelfand-Shilov spaces

We study continuity of the multiplier operator $e^{i q}$ acting on Gelfand--Shilov spaces, where $q$ is a polynomial on $\mathbf R^d$ of degree at least two with real coefficients. In the parameter quadrant for the spaces we identify a wedge that depends on the polynomial degree for which the operator is continuous. We also show that in a large part of the complement region the operator is not continuous in dimension one. The results give information on well-posedness for linear evolution equations that generalize the Schrödinger equation for the free particle.

math.FA↗
arXiv2024

Schwartz very weak solutions for Schr\"odinger type equations with distributional coefficients

This paper continues the analysis of Schr\"odinger type equations with distributional coefficients initiated by the authors in [3]. Here we consider coefficients that are tempered distributions with respect to the space variable and are continuous in time. We prove that the corresponding Cauchy problem, which in general cannot even be stated in the standard distributional setting, admits a Schwartz very weak solution which is unique modulo negligible perturbations. Consistency with the classical theory is proved in the case of regular coefficients and Schwartz Cauchy data.

math.AP↗
arXiv2024

$L^p-L^q$ estimates for solutions to the plate equation with mass term

In this paper, we study the Cauchy problem for the linear plate equation with mass term and its applications to semilinear models. For the linear problem we obtain $L^p-L^q$ estimates for the solutions in the full range $1\leq p\leq q\leq \infty$, and we show that such estimates are optimal. In the sequel, we discuss the global in time existence of solutions to the associated semilinear problem with power nonlinearity $|u|^α$. For low dimension space $n\leq 4$, and assuming $L^1$ regularity on the second datum, we were able to prove global existence for $α> \max\{α_c(n), \tildeα_c(n)\}$ where $α_c = 1+4/n$ and $\tilde α_c = 2+2/n$. However, assuming initial data in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$, the presence of the mass term allows us to obtain global in time existence for all $1<α\leq (n+4)/[n-4]_+$. We also show that the latter upper bound is optimal, since we prove that there exist data such that a non-existence result for local weak solutions holds when $α> (n+4)/[n-4]_+$.

math.AP↗
arXiv2024

Gevrey well posedness for $3$-evolution equations with variable coefficients

We study the Cauchy problem for a class of third order linear anisotropic evolution equations with complex valued lower order terms depending both on time and space variables. Under suitable decay assumptions for $|x| \to \infty$ on these coefficients, we prove a well posedness result in Gevrey-type spaces.

math.AP↗
arXiv2024

Schrödinger type equations with singular coefficients and lower order terms

In this paper we investigate the well-posedness of the Cauchy problem for a Schrödinger operator with singular lower order terms. We allow distributional coefficients and we approach this problem via the regularising methods at the core of the theory of very weak solutions. We prove that a very weak solution exists and it is unique modulo negligible perturbations. Very weak solutions converge to classical solutions when the equation coefficients are regular enough.

math.AP↗
arXiv2022

KdV-type equations in projective Gevrey classes

We prove well-posedness of the Cauchy problem for a class of third order quasilinear evolution equations with variable coefficients in projective Gevrey spaces. The class considered is connected with several equations in Mathematical Physics as the KdV and KdVB equation and some of their many generalizations.

math.AP↗
arXiv2021

The Cauchy problem for $3$-evolution equations with data in Gelfand-Shilov spaces

We consider the Cauchy problem for a $3$-evolution operator $P$ with $(t,x)$-depending coefficients and complex valued lower order terms. We assume the initial data to be Gevrey regular and to admit an exponential decay at infinity, that is, the data belong to some Gelfand-Shilov spaces of type $\mathscr{S}$. Under suitable assumptions on the decay at infinity of the imaginary parts of the coefficients of $P$ we prove the existence of a solution with the same Gevrey regularity of the data and we describe its behavior for $|x| \to\infty$.

math.AP↗
arXiv2018

Global Gevrey hypoellipticity on the torus for a class of systems of complex vector fields

Let $L_j = \partial_{t_j} + (a_j+ib_j)(t_j) \partial_x, \, j = 1, \dots, n,$ be a system of vector fields defined on the torus $\mathbb{T}_t^{n}\times\mathbb{T}_x^1$, where the coefficients $a_j$ and $b_j$ are real-valued functions belonging to the Gevrey class $G^s(\mathbb{T}^1)$, with $s>1$. In this paper we were able to characterize the global $s-$hypoellipticity of this system in terms of Diophantine approximations and the Nirenberg-Treves condition (P).

math.AP↗