相关论文: Weighted estimates and Fujita exponent for a nonlo…

We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho…

This paper explores the critical behavior of the semilinear heat equation $u_t+\mathcal{L}_{a, b}u=|u|^p+f(x)$, considering both the presence and absence of a forcing term $f(x).$ The mixed local-nonlocal operator $\mathcal{L}_{a,…

In this paper we consider the blow-up problem for a mixed local-nonlocal diffusion operator, \[ u_t=a\Delta u -b(-\Delta)^s u+u^p. \] We show that the Fujita exponent is given by the nonlocal part, $p_F=1+2s/N$. We also determinate, in some…

This paper concerns the Cauchy problems for the nonlinear Rayleigh-Stokes equation and the corresponding system with time-fractional derivative of order $\alpha\in(0,1)$, which can be used to simulate the anomalous diffusion in viscoelastic…

We consider non-negative solutions to the semilinear space-fractional diffusion problem $(\partial_t+(-\Delta)^{\alpha/2})u=\rho(x)u^p$ on whole space $R^n$ with nonnegative initial data and with $(-\Delta)^{\alpha/2}$ being the…

We prove the boundedness of the non-local operator \[ \mathcal{L}^a u(x)=\int_{\mathbb{R}^d} \left(u(x+y)-u(x)-\chi_\alpha(y)\big(\nabla u(x),y\big)\right) a(x,y)\frac{dy}{|y|^{d+\alpha}} \] from $H_{p,w}^\alpha(\mathbb{R}^d)$ to…

In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form $$\begin{cases} &u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\…

We consider nonlocal operators of the form \begin{equation*} L_t u(x) = \int_{\mathbb{R}^d} \left( u(x+y)-u(x)-\nabla u(x)\cdot y^{(\sigma)} \right) \nu_t(dy), \end{equation*} where $\nu_t$ is a general L\'evy measure of order $\sigma…

We investigate the Cauchy problem for a semilinear spatio--temporal fractional diffusion equation with a time-dependent forcing term: \[ \partial_t^\alpha u + (-\Delta)^{\mathsf{s}} u = |u|^p + t^{\sigma}\,\mathbf{w}(x), \quad (t,x) \in…

We look at estimates for the Green's function of time-fractional evolution equations of the form $D^{\nu}_{0+*} u = Lu$, where $D^{\nu}_{0+*}$ is a Caputo-type time-fractional derivative, depending on a L\'evy kernel $\nu$ with variable…

We investigate the global continuity on $L^p$ spaces with $p\in [1,\infty]$ of Fourier integral operators with smooth and rough amplitudes and/or phase functions subject to certain non-degeneracy conditions. We initiate the investigation of…

We consider the Cauchy problem for a class of non-linear evolution equations in the form \[L(\partial_t,\partial_x) u=F(\partial_t^\ell u), \quad (t,x)\in [0,\infty)\times \mathbb{R}^n;\] here, $L(\partial_t,\partial_x)$ is a linear partial…

We develop a weighted mixed-norm $L_q(L_p)$-estimates for solutions to fractional evolution equations of the form \[ \partial_t^\alpha w(t,x) = \phi(\Delta) w(t,x) + h(t,x), \quad w(0,\cdot) = w_0, \quad t > 0, \; x \in \mathbb{R}^d, \]…

In this paper we first study the generalized weighted Hardy spaces $H^p_{L,w}(X)$ for $0<p\le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of…

We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ \partial_t^\alpha u - Lu + \lambda u= f \quad \mathrm{in} \quad (0,T) \times \mathbb{R}^d,$$ where $\partial_t^\alpha u$ is the Caputo…

We prove weighted estimates for singular integral operators which operate on function spaces on a half-line. The class of admissible weights includes Muckenhoupt weights and weights satisfying Sawyer's one-sided conditions. The kernels of…

We are concerned with the Cauchy problem for the semilinear parabolic equation driven by the mixed local-nonlocal operator $\mathcal{L} = -\Delta+(-\Delta)^s$, with a power-like source term. We show that the so-called Fujita phenomenon…

This paper is concerned with the existence/nonexistence of nontrivial global-in-time solutions to the Cauchy problem \begin{equation} \begin{cases}\tag{P}\partial_tu-\partial_x^2u+Vu=(1+x^2)^{-\frac{m}{2}}u^p,&x\in\mathbb{R},\ t>0,\\…

In this paper, we establish Schauder's estimates for the following non-local equations in \mR^d : $$ \partial_tu=\mathscr L^{(\alpha)}_{\kappa,\sigma} u+b\cdot\nabla u+f,\ u(0)=0, $$ where $\alpha\in(1/2,2)$ and $ b:\mathbb R_+\times\mathbb…

The behaviour of solutions for a non-linear diffusion problem is studied. A subordination principle is applied to obtain the variation of parameters formula in the sense of Volterra equations, which leads to the integral representation of a…