arXiv subjects

Arlúcio Viana

Publications and source records attributed to Arlúcio Viana.

5 recordsLinked to original sources

arXiv2026

On semilinear Grushin--Schrödinger equation in $\mathbb{R}^N$

We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -Δ_γu + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N, \end{equation*} where $Δ_γ$ denotes the so-called Grushin-type operator in $\mathbb{R}^N$. The potentials $V$ and $Q$ are assumed to be controlled below and above, respectively, by functions of type $(1+|z|)^a$, $a\in\mathbb{R}$. The main result is the embedded of the space $E_V^γ$ into the weighted Lebesgue space $L_Q^p(\mathbb{R}^N)$, under suitable conditions. Finally, we derive regularity results for the obtained weak solutions.

math.AP↗

arXiv2025

Nonexistence of global solutions to the Grushin heat equation with nonlocal and local nonlinearities

In this paper, we investigate the nonexistence of global solutions to the Grushin-type heat equation with nonlinear reaction terms, including cases involving memory effects: $$ \left\{\begin{array}{ll} \displaystyle {u_{t}-Δ_{\mathcal{G}} u = k_1 \int_0^t(t-s)^{-γ}\abs{u}^{p_1-1}u(s)\,\mathrm{d}s} + k_2|u|^{p_2-1}u, & (z,t)\in {\mathbb{R}}^{N+k}\times (0,\infty), \displaystyle{u(z,0)= u_0(z),\qquad\qquad}&\displaystyle{z\in {\mathbb{R}}^{N+k},} \end{array} \right. $$ where $Δ_{\mathcal{G}}$ denotes the Grushin operator, $u_0 \in L^1_{\mathrm{loc}}(\mathbb{R}^{N+k})$, $γ\in[0,1)$, $k_1,k_2 \geq 0$, and $p_1,p_2>1$. We establish sharp nonexistence results for global-in-time positive solutions, thereby completing the picture of global existence versus blow-up and allow us to identify the corresponding Fujita-type critical exponents in certain parameter regimes. The analysis relies on the test function method, adapted to handle both the degeneracy of the Grushin operator and the influence of the memory term.

math.AP↗

arXiv2025

Duality estimates for subdiffusion problems including time-fractional porous medium type equations

We prove duality estimates for time-fractional and more general subdiffusion problems. An important example is given by subdiffusive porous medium type equations. Our estimates can be used to prove uniqueness of weak solutions to such problems, and they allow to extend a key estimate from classical reaction-diffusion systems to the subdiffusive case. Besides concrete equations involving a Laplacian, we also consider abstract problems in a Hilbert space setting.

math.AP↗

arXiv2025

On the semilinear heat equation with the Grushin operator

In this work, we study the heat equation with Grushin's operator. We present an expression for its heat kernel, prove its decay in $L^p$ spaces, and that it is an approximation of the identity. As a consequence, the heat semigroup associated with Grushin's operator using this heat kernel is estimated in Lebesgue spaces. Next, we use the results to prove the existence, uniqueness, continuous dependence, and blowup alternative of mild solutions of a nonlinear Cauchy problem associated with Grushin's operator. A global existence result is also presented.

math.AP↗

arXiv2025

On the Heat equation involving a Grushin operator in Marcinkiewcz spaces

In this work, we give sufficient conditions for the existence and uniqueness of the heat equation involving the operator $$ Δ_{\mathcal{G}}=\dfrac{1}{2}\left(Δ_{x}+|x|^2Δ_{y}\right) $$ in Marcinkiewicz spaces. Furthermore, we provide sufficient conditions for the existence of positive, symmetric, and self-similar solutions.

math.AP↗