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Ahmad Z. Fino

Publications and source records attributed to Ahmad Z. Fino.

At least 19 recordsLinked to original sources
arXiv2026

Heat equations driven by mixed local-nonlocal operators with exponential nonlinearity

We investigate the Cauchy problem for a heat equation driven by the mixed local-nonlocal operator $\mathcal{L}:=-Δ+(-Δ)^s$, $s\in(0,1)$, with exponential nonlinearity \[ \partial_tu(x,t)+\mathcal{L}u(x,t)=f(u(x,t)), \qquad (x,t)\in \mathbb{R}^{d}\times(0,\infty), \] where $f:\mathbb{R}\to\mathbb{R}$ exhibits exponential growth at infinity and satisfies $f(0)=0$. We establish local well-posedness in a suitable Orlicz space in the case where $f(u)\sim e^{|u|^p}$ as $|u|\to\infty$, with $p>1$. We further prove the existence of global solutions for small initial data under the assumption that $f$ satisfies the growth condition $|f(u)|\sim |u|^m$ near the origin. Moreover, we derive large-time decay estimates in Lebesgue spaces, showing that the behavior of the nonlinearity near the origin determines the decay rate of solutions and highlights a unique asymptotic transition that bridges local and non-local diffusion theories.

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arXiv2026

Parabolic problems whose Fujita critical exponent is not given by scaling

This paper investigates the (fractional) heat equation with a nonlocal nonlinearity involving a Riesz potential: \begin{equation*} u_{t}+(-Δ)^{\fracβ{2}} u= I_α(|u|^{p}),\qquad x\in \mathbb{R}^n,\,\,\,t>0, \end{equation*} where $α\in(0,n)$, $β\in(0,2]$, $n\geq1$, $p>1.$ We introduce the Fujita-type critical exponent $p_{\mathrm{Fuj}}(n,β,α)=1+(β+α)/(n-α)$, which characterizes the global behavior of solutions: global existence for small initial data when $p>p_{\mathrm{Fuj}}(n,β,α),$ and finite-time blow-up when $p\leq p_{\mathrm{Fuj}}(n,β,α)$. It is remarkable that the critical Fujita exponent is not determined by the usual scaling argument that yields $p_{sc}=1+(β+α)/n$, but instead arises in an unconventional manner, similar to the results of Cazenave et al. [Nonlinear Analysis, 68 (2008), 862-874] for the heat equation with a nonlocal nonlinearity of the form $\int_0^t(t-s)^{-γ}|u(s)|^{p-1}u(s)ds,\,0\leq γ<1.$ The result on global existence for $p>p_{\mathrm{Fuj}}(n,2,α),$ provides a positive answer to the hypothesis proposed by Mitidieri and Pohozaev in [Proc. Steklov Inst. Math., 248 (2005) 164-185]. We further establish global nonexistence results for the above heat equation, where the Riesz potential term $I_α(|u|^{p})$ is replaced by a more general convolution operator $(\mathcal{K}\ast |u|^p),\,\mathcal{K}\in L^1_{loc}$, thereby extending the Mitidieri-Pohozaev's results established in the aforementioned work. Proofs of the blow-up results are obtained using a nonlinear capacity method specifically adapted to the structure of the problem, while global existence is established via a fixed-point argument combined with the Hardy-Littlewood-Sobolev inequality.

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arXiv2025

The Fujita exponent for a heat equation with mixed local and nonlocal nonlinearities on the Heisenberg group

This article deals with the problems of local and global solvability for a semilinear heat equation on the Heisenberg group involving a mixed local and nonlocal nonlinearity. The characteristic features of such equations, arising from the interplay between the geometric structure of the Heisenberg group and the combined nonlinearity, are analyzed in detail. The need to distinguish between subcritical and supercritical regimes is identified and justified through rigorous analysis. On the basis of the study, the author suggests precise conditions under which local-in-time mild solutions exist uniquely for regular, nonnegative initial data. It is proved that global existence holds under appropriate growth restrictions on the nonlinear terms. To complement these results, it is shown, by employing the capacity method, that solutions cannot exist globally in time when the nonlinearity exceeds a critical threshold. As a result, the Fujita exponent is formulated and identified as the dividing line between global existence and finite-time blow-up. In addition, lifespan estimates were obtained in the supercritical regime, providing insight into how the size of the initial data influences the time of blow-up.

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arXiv2025

Fujita-type results for parabolic equations with Hartree-type nonlinearities

This paper investigates the critical behavior of global solutions to a parabolic equation with a Hartree-type nonlinearity of the form $$\left\{\begin{array}{ll} u_{t}+(-Δ)^{\fracβ{2}} u= (\mathcal{K}\ast |u|^{p})|u|^{q},&\qquad x\in \mathbb{R}^n,\,\,\,t>0, u(x,0)=u_{0}(x),& \qquad x\in \mathbb{R}^n,\end{array} \right.$$ where $β\in(0,2]$, $n\geq1$, $p>1$, $q\geq 1$, $(-Δ)^{\fracβ{2}},\,β\in(0,2)$ denotes the fractional Laplacian, the symbol $\ast$ denotes the convolution operation in $\mathbb{R}^n$, and $\mathcal{K}:(0,\infty)\rightarrow(0,\infty)$ is a continuous function such that $\mathcal{K}(|\cdotp|)\in L^1_{loc}(\mathbb{R}^n)$ and is monotonically decreasing in a neighborhood of infinity. We establish conditions for the global nonexistence of solutions to the problem under consideration, thereby partially improving some results of Filippucci and Ghergu in [Discrete Contin. Dyn. Syst. A, 42 (2022) 1817-1833] and [Nonlinear Anal., 221 (2022) 112881]. In addition, we establish local and global existence results in the case where the convolution term corresponds to the Riesz potential. Our methodology relies on the nonlinear capacity method and the fixed-point principle, combined with the Hardy-Littlewood-Sobolev inequality.

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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.

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arXiv2025

Critical exponent for the one-dimensional wave equation with a space-dependent scale invariant damping and time derivative nonlinearity

We investigate in this paper the Cauchy problem of the one-dimensional wave equation with space-dependent damping of the form $μ_0(1+x^2)^{-1/2}$, where $μ_0>0$, and time derivative nonlinearity. We establish global existence of mild solutions for small data compactly supported by employing energy estimates within suitable Sobolev spaces of the associated homogeneous problem. Furthermore, we derive a blow-up result under some positive initial data by employing the test function method. This shows that the critical exponent is given by $p_G(1+μ_0)=1+2/μ_0$, when $μ_0\in (0,1]$, where $p_G$ is the Glassey exponent. To the best of our knowledge, this constitutes the first identification of the critical exponent range for this class of equations. As by product, we extend the global existence result to a more general class of space/time nonlinearities of the form $f(\partial_tu,\partial_x u)=|\partial_x u|^{q}$ or $f(\partial_tu,\partial_x u)=|\partial_tu|^{p}|\partial_x u|^{q}$, with $p,q>1$.

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arXiv2025

Local and global solvability of the Grushin heat equation with mixed nonlinear memory and reaction terms

In this work, we investigate the solvability of a heat equation involving the Grushin operator. The equation is perturbed by two nonlinear reaction terms, one of which includes a memory component, introducing nonlocal effects in time. We first establish local well-posedness results in both Lebesgue spaces and in the space of continuous functions vanishing at infinity. Furthermore, we develop and apply comparison principles that allow us to derive sufficient conditions for the global existence of solutions, depending on the relative strength and nature of the nonlinearities involved. In particular, we highlight how the presence of the memory term influences global solvability. Our results contribute to the understanding of parabolic equations with degenerate operators and complex nonlinear interactions.

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arXiv2025

Global existence for wave equations with scale-invariant time-dependent damping and time derivative nonlinearity

This paper addresses the Cauchy problem for wave equations with scale-invariant time-dependent damping and nonlinear time-derivative terms, modeled as $$\partial_{t}^2u- Δu +\fracμ{1+t}\partial_tu= f(\partial_tu), \quad x\in \mathbb{R}^n, t>0,$$ where $f(\partial_tu)=|\partial_tu|^p $ or $|\partial_tu|^{p-1}\partial_tu$ with $p>1$ and $μ>0$. We prove global existence of small data solutions in low dimensions $1\leq n\leq 3$ by using energy estimates in appropriate Sobolev spaces. Our primary contribution is an existence result for $p>1+\frac2μ$, in the one-dimensional case, when $μ\le 2$, which in conjunction with prior blow-up results from \cite{Our2}, establish that the critical exponent for small data solutions in one dimension is $p_G(1,μ)=1+\frac2μ$, when $μ\le 2$. To the best of our knowledge, this is the first identification of the critical exponent range for the time-dependent damped wave equations with scale-invariant and time-derivative nonlinearity.

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arXiv2025

Cazenave-Dickstein-Weissler-type extension of Fujita's problem on Heisenberg groups

This paper examines the critical exponents for the existence of global solutions to the equation \begin{equation*} \begin{array}{ll} \displaystyle u_t-Δ_{\mathbb{H}}u=\int_0^t(t-s)^{-γ}|u(s)|^{p-1}u(s)\,ds,&\qquad 0\leqγ<1,\,\,\, {η\in \mathbb{H}^n,\,\,\,t>0,} \end{array}\end{equation*} on the Heisenberg groups $\mathbb{H}^n.$ There exists a critical exponent $$p_c= \max\Big\{\frac{1}γ,p_γ\Big\}\in(0,+\infty],\quad\hbox{with}\quad p_γ=1+\frac{2(2-γ)}{Q-2+2γ},\,\,Q=2n+2$$ such that for all $1 p_c$, a global positive solution exists if the initial data is sufficiently small. The results obtained are a natural extension of the results of Cazenave et al. [Nonlinear Analysis 68 (2008), 862-874], where similar studies were carried out in $\mathbb{R}^n$. Also given are several theorems concerning the lifespan estimates of local solutions for different cases of initial data. The proofs of the main results are based on test function methods and Banach fixed point principle.

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arXiv2025

Decay of mass for a semilinear heat equation on Heisenberg group

In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation with time-dependent absorption $u_{t}-Δ_{\mathbb{H}}u=- k(t)u^p$ posed on $\mathbb{H}^n$, driven by the Heisenberg Laplacian and supplemented with a nonnegative integrable initial data, where $p>1$, $n\geq 1$, and $k:(0,\infty)\to(0,\infty)$ is a locally integrable function. We study the large time behavior of non-negative solutions and show that the nonlinear term determines the large time asymptotic for $p\leq 1+2/Q,$ while the classical/anomalous diffusion effects win if $p>1+{2}/{Q}$, where $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$.

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arXiv2025

Decay of mass for a semilinear heat equation with mixed local-nonlocal operators

In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation $\partial_t u+t^β\mathcal{L} u= - h(t)u^p$ posed on $\mathbb{R}^N$, driven by the mixed local-nonlocal operator $\mathcal{L}=-Δ+(-Δ)^{α/2}$, $α\in(0,2)$, and supplemented with a nonnegative integrable initial data, where $p>1$, $β\geq 0$, and $h:(0,\infty)\to(0,\infty)$ is a locally integrable function. We study the large time behavior of non-negative solutions and show that the nonlinear term determines the large time asymptotic for $p\leq 1+α/{N(β+1)},$ while the classical/anomalous diffusion effects win if $p>1+α/{N(β+1)}$.

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arXiv2024

Fujita type results for a parabolic inequality with a non-linear convolution term on the Heisenberg group

The purpose of this paper is to investigate the non-existence of global weak solutions of the following degenerate inequality on the Heisenberg group $$ \begin{cases} u_{t}-Δ_{\mathbb{H}}u\geq (\mathcal{K}\ast_{_{\mathbb{H}}}|u|^p)|u|^q ,\qquad {η\in \mathbb{H}^n,\,\,\,t>0,} \\{}\\ u(η,0)=u_{0}(η), \qquad\qquad\qquad\quad η\in \mathbb{H}^n, \end{cases} $$ where $n\geq1$, $p,q>0$, $u_0\in L^1_{loc}(\mathbb{H}^n)$, $Δ_{\mathbb{H}}$ is the Heisenberg Laplacian, and $\mathcal{K}:(0,\infty)\rightarrow(0,\infty)$ is a continuous function satisfying $\mathcal{K}(|\cdotp|_{_{\mathbb{H}}})\in L^1_{loc}(\mathbb{H}^n)$ which decreases in a vicinity of infinity. In addition, $\ast_{_{\mathbb{H}}}$ denotes the convolution operation in $\mathbb{H}^n$. Our approach is based on the non-linear capacity method.

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arXiv2023

Fujita-type results for the degenerate parabolic equations on the Heisenberg groups

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups.

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arXiv2022

Some nonexistence results for space-time fractional Schr{ö}dinger equations without gauge invariance

In this paper, we consider the Cauchy problem in $\mathbb{R}^N$, $N\geq1$, for semi-linear Schrödinger equations with space-time fractional derivatives. We discuss the nonexistence of global $L^1$ or $L^2$ weak solutions in the subcritical and critical cases under some conditions on the initial data and the nonlinear term. Furthermore, the nonexistence of local $L^1$ or $L^2$ weak solutions in the supercritical case are studied.

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arXiv2022

Blow-up and lifespan estimate for wave equations with critical damping term of space-dependent type related to Glassey conjecture

The main purpose of the present paper is to study the blow-up problem of the wave equation with space-dependent damping in the \textit{scale-invariant case} and time derivative nonlinearity with small initial data. Under appropriate initial data which are compactly supported, by using a test function method and taking into account the effect of the damping term ($\fracμ{\sqrt{1+|x|^2}}u_t$), we provide that in higher dimensions the blow-up region is given by $p \in (1, p_G(N+μ)]$ where $p_G(N)$ is the Glassey exponent. Furthermore, we shall establish a blow-up region, independent of $μ$ given by $p\in (1, 1+\frac{2}{N}),$ for appropriate initial data in the energy space with noncompact support.

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arXiv2022

A competition on blow-up for semilinear wave equations with scale-invariant damping and nonlinear memory term

In this paper, we investigate blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term in $\mathbb{R}^n$, which can be represented by the Riemann-Liouville fractional integral of order $1-γ$ with $γ\in(0,1)$. Our main interest is to study mixed influence from damping term and the memory kernel on blow-up conditions for the power of nonlinearity, by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of $γ$, on the blow-up range between the effective case and the non-effective case.

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arXiv2021

Blow-up of solutions to semilinear wave equations with a time-dependent strong damping

The paper investigates a class of a semilinear wave equation with time-dependent damping term ($-\frac{1}{{(1+t)}^β}Δu_t$) and a nonlinearity $|u|^p$. We will show the influence of the the parameter $β$ in the blow-up results under some hypothesis on the initial data and the exponent $p$ by using the test function method. We also study the local existence in time of mild solution in the energy space $H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$.

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