arXiv subjects
Publications and source records attributed to Miguel Loayza.
In this work, we establish optimal conditions concerning the global and nonglobal existence of solutions of a semilinear parabolic equations governed by a mixed local-nonlocal operator. Furthermore, our findings recover the Fujita exponent recently derived by Biagi, Punzo and Vecchi, as well as by Del Pezzo and Ferreira.
We are concerned with the existence of global and blow-up solutions for the nonlinear parabolic problem described by the Hardy-Hénon equation $u_t - Δ_{\mathbb{H}} u = |\cdot|_{\mathbb{H}}^γ u^p \mbox{ in } \mathbb{H}^N \times (0,T),$ where $\mathbb{H}^N$ is the $N$-dimensional Heisenberg group, and the singular term $|\cdot|_{\mathbb{H}}^γ$ is given by the Korányi norm. Our study focuses on nonnegative solutions. We establish that for $γ\geq 0$, the Fujita critical exponent is $p_c = 1+ (2+γ)/Q$, where $Q=2N+2$ is the homogeneous dimension of $\mathbb{H}^N$. For $γ<0$, the solutions blow up for $1 1+ (2+γ)/(Q + γ)$. In particular, our results coincide with the results found by Georgiev and Palmieri in \cite{PALMIERI} for $γ=0$.
We establish both the existence and uniqueness of non-negative global solutions for the nonlinear heat equation $u_t-Δu=|x|^{-γ}\,u^q$, $0 0$ in the whole space $\mathbb{R}^N$, and for non-negative initial data $u_0\in C_0(\mathbb{R}^N)$.
In this paper, we obtain the so-called Fujita exponent to the following parabolic system with time-weighted sources and degenerate coefficients $ u_{t}- \mbox{div} ( ω(x)\nabla { u} )= t^{r} v^{p} $ and $ v_{t}- \mbox{div} ( ω(x)\nabla {v} )= t^{s} u^{p}$ in $\mathbb{R}^{N} \times (0,T)$ with initial data belonging to $ \left[L^\infty(\mathbb{R}^N)\right]^2.$ Where $p,q > 0$ with $ pq > 1$; $r,s>-1 $; and either $ω(x) = | x_1|^{a},$ or $ω(x) = | x |^{b}$ with $a,b > 0$.
We provide new conditions for the local existence of solutions to the time-weighted parabolic equation $ u_t - Δu = h(t)f(u) \mbox{ in } Ω\times (0,T),$ where $ Ω$ is a arbitrary smooth domain, $f\in C(\mathbb{R})$, $h\in C([0,\infty))$ and $u(0)\in L^r(Ω)$. As consequence of our results, considering a suitable behavior of the non-negative initial data, we obtain a second critical value $ρ^\star = 2r/(p-1),$ when $f(u)=u^p$ and $p> 1 + 2r/N$, which determines the existence (or not) of a local solution $u \in L^\infty((0,T), L^r(Ω)).$
We consider the cauchy problem for the fractional power dissipative equation $u_t+(-Δ)^{β/2} u=F(u)$, where $β>0$ and $F(u)=B(u, ...,u)$ and $B$ is a multilinear form on a Banach space $E$. We show a global existence result assuming some properties of scaling degree of the multilinear form and the norm of the space $E$. We extend the ideas used for the treating of the equation to determine the global existence for the system $u_t+(-Δ)^{β/2}= F(v)$, $v_t+(-Δ)^{β/2}v=G(u)$ where $F(u)=B_1(u,...,u), G(v)=B_2(v,...,v)$