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Register a new userOn the existence and uniqueness of weak solutions to time-fractional elliptic equations with time-dependent variable coefficients
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Authors
H.T. Tuan
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This paper is devoted to discussing the existence and uniqueness of weak solutions to time-fractional elliptic equations having time-dependent variable coefficients. To obtain the main result, our strategy is to combine the Galerkin method, a basic inequality for the fractional derivative of convex Lyapunov candidate functions, the Yoshida approximation sequence and the weak compactness argument.
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We prove uniqueness for weak solutions to abstract parabolic equations with the fractional Marchaud or Caputo time derivative. We consider weak solutions in time for divergence form equations when the fractional derivative is transferred to the test function.
We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to zero or to infinity.
We investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined among other coefficients of these equations, by the knowledge of a suitable time-sequence of partial Dirichlet-to-Neumann maps.
This work is the second of the series of three papers devoted to the study of asymptotic dynamics in the chemotaxis system with space and time dependent logistic source,$$partial_tu=Delta u-chi ablacdot(u abla v)+u(a(x,t)-ub(x,t)),quad 0=Delta v-lambda v+mu u , xinmathbb{R}^N, (1) $$where $Nge1$ is a positive integer, $chi,lambda,mu>0$, and the functions $a(x,t), b(x,t)$ are positive and bounded. In the first of the series, we studied the phenomena of pointwise and uniform persistence, and asymptotic spreading for solutions with compactly supported or front like initials. In the second of the series, we investigate the existence, uniqueness and stability of strictly positive entire solutions. In this direction, we prove that, if $0lemuchi<b_inf$, then (1) has a strictly positive entire solution, which is time-periodic (respectively time homogeneous) when the logistic source function is time-periodic (respectively time homogeneous). Next, we show that there is positive constant $chi_0$, such that for every $0lechi<chi_0$, (1) has a unique positive entire solution which is uniform and exponentially stable with respect to strictly positive perturbations. In particular, we prove that $chi_0$ can be taken to be $b_inf/2mu$ when the logistic source function is either space homogeneous or the function $b(x,t)/a(x,t)$ is constant. We also investigate the disturbances to Fisher-KKP dynamics caused by chemotatic effects, and prove that$$sup_{0<chilechi_1}sup_{t_0inmathbb{R},tge 0}frac{1}{chi}|u_{chi}(cdot,t+t_0;t_0,u_0)-u_0(cdot,t+t_0;t_0,u_0)|_{infty}<infty$$for every $0<chi_1<b_inf/mu$ and every uniformly continuous initial function $u_0$, with $u_{0inf}>0$, where $(u_chi(x,t+t_0;t_0,u_0),v_chi(x,t+t_0;t_0,u_0))$ denotes the unique classical solution of (1) with $u_chi(x,t_0;t_0,u_0)=u_0(x)$, for every $0lechi<b_inf$.
We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $Omega$ in $mathbb{R}^n$, $n ge 3$, with drifts $mathbf{b}$ in the critical weak $L^n$-space $L^{n,infty}(Omega ; mathbb{R}^n )$. First, assuming that the drift $mathbf{b}$ has nonnegative weak divergence in $L^{n/2, infty }(Omega )$, we establish existence and uniqueness of weak solutions in $W^{1,p}(Omega )$ or $D^{1,p}(Omega )$ for any $p$ with $n = n/(n-1)< p < n$. By duality, a similar result also holds for the dual problem. Next, we prove $W^{1,n+varepsilon}$ or $W^{2, n/2+delta}$-regularity of weak solutions of the dual problem for some $varepsilon, delta >0$ when the domain $Omega$ is bounded. By duality, these results enable us to obtain a quite general uniqueness result as well as an existence result for weak solutions belonging to $bigcap_{p< n }W^{1,p}(Omega )$. Finally, we prove a uniqueness result for exterior problems, which implies in particular that (very weak) solutions are unique in both $L^{n/(n-2),infty}(Omega )$ and $L^{n,infty}(Omega )$.
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