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Register a new userGlobal existence and spatial analyticity for a nonlocal flux with fractional diffusion
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In this paper, we study a one dimensional nonlinear equation with diffusion $- u(-partial_{xx})^{frac{alpha}{2}}$ for $0leq alphaleq 2$ and $ u>0$. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space $L^1(mathbb{R})cap H^{1/2}(mathbb{R})$ when $0leqalphaleq 2$. For subcritical $1<alphaleq 2$ and critical case $alpha=1$, we obtain global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrap method to improve the regularity of mild solutions in Bessel potential spaces for subcritical case $1<alphaleq 2$. Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case $alpha=1$, if the initial data $rho_0$ satisfies $- u<infrho_0<0$, we use the characteristics methods for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If $rho_0geq0$, the solution exists globally and converges to steady state.
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In this paper we address the analytical investigation of a model for adhesive contact, which includes nonlocal sources of damage on the contact surface, such as the elongation. The resulting PDE system features various nonlinearities rendering the unilateral contact conditions, the physical constraints on the internal variables, as well as the integral contributions related to the nonlocal forces. For the associated initial-boundary value problem we obtain a global-in-time existence result by proving the existence of a local solution via a suitable approximation procedure and then by extending the local solution to a global one by a nonstandard prolongation argument.
In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $mathbb{R}^d$ and a complete Riemannian manifold $mathrm{M}$. On one hand, in $mathbb{R}^d$, we prove that any solution $u=u(t,x)$ to $u_t(t,x)-mathrm{L}_alpha^{kappa} u(t,x)=0$, where $mathrm{L}_alpha^{kappa}$ is a nonlocal operator of order $alpha$, is time analytic in $(0,1]$ if $u$ satisfies the growth condition $|u(t,x)|leq C(1+|x|)^{alpha-epsilon}$ for any $(t,x)in (0,1]times mathbb{R}^d$ and $epsilonin(0,alpha)$. We also obtain pointwise estimates for $partial_t^kp_alpha(t,x;y)$, where $p_alpha(t,x;y)$ is the fractional heat kernel. Furthermore, under the same growth condition, we show that the mild solution is the unique solution. On the other hand, in a manifold $mathrm{M}$, we also prove the time analyticity of the mild solution under the same growth condition and the time analyticity of the fractional heat kernel, when $mathrm{M}$ satisfies the Poincare inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in $mathbb{R}^d$ using the method of Fourier transform and contour integrals. We find that when $alphain (0,1]$, the fractional heat kernel is time analytic at $t=0$ when $x eq 0$, which differs from the standard heat kernel. As corollaries, we obtain sharp solvability condition for the backward fractional heat equation and time analyticity of some nonlinear fractional heat equations with power nonlinearity of order $p$. These results are related to those in [8] and [11] which deal with local equations.
This paper deals with a boundary-value problem in three-dimensional smooth bounded convex domains for the coupled chemotaxis-Stokes system with slow $p$-Laplacian diffusion begin{equation} onumber left{ begin{aligned} &n_t+ucdot abla n= ablacdotleft(| abla n|^{p-2} abla nright)- ablacdot(n abla c), &xinOmega, t>0, &c_t+ucdot abla c=Delta c-nc,&xinOmega, t>0, &u_t=Delta u+ abla P+n ablaphi ,&xinOmega, t>0, & ablacdot u=0, &xinOmega, t>0, end{aligned} right. end{equation} where $phiin W^{2,infty}(Omega)$ is the gravitational potential. It is proved that global bounded weak solutions exist whenever $p>frac{23}{11}$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0geq 0$ and $c_0geq 0$.
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.
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