We show the following equivalence of solutions to general fractional Helmholtz equations: $u$ is a solution to $$A^su=\lambda^su$$ for \textit{some} fixed $00$ if and only if $u$ is a solution to the same equation for \textit{all} $s\neq0$. Furthermore, we show as well that $u$ equivalently solves the logarithmic Helmholtz equation $$\log(A)u=(\log\lambda)u.$$ Here, $A$ is a nonnegative, linear operator on a Banach space $X$. In particular, our results hold whenever $A=-L$, where $L$ is the infinitesimal generator of a $C_0$-semigroup in a Banach space $X$ satisfying mild assumptions that are typical in applications. As a particular case, we recover known results for the fractional Laplacian $(-\Delta)^s$ in $\mathbb{R}^n$. More importantly, we provide a list of examples of other fractional power operators in settings where the Fourier transform is not available and for which our theorems apply. These include fractional powers and logarithms of second order elliptic operators in bounded domains, Laplace--Beltrami operators on Riemannian manifolds, space-time master equations, the discrete Laplacian, and fractional derivatives.
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
We establish sharp temporal lower bounds for the full spatial $L^2$-norm of energy solutions to $\partial_tu-\Delta u=vu$ on $\mathbb{R}^n$, with $n\geq3$. For global solutions, nonvanishing at a single time implies the lower bound $ce^{-Ct}$ when $v$ is small in the scale-invariant space $L^\infty_tL^{n/2}_x$, and $ce...
We study the equations $\Delta_p^H u=u^{-q}$, $u>0$, and $-\Delta_p^H u=e^u$ in $\mathbb{R}^N$, where $\Delta_p^H u=\operatorname{div}(H(\nabla u)^{p-1}\nabla H(\nabla u))$ is the Finsler $p$-Laplacian associated with a uniformly convex norm $H$, and $p\ge2$. We prove nonexistence of solutions stable outside a compact...
We study stable solutions of the mean curvature equation $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) =|u|^{p-1}u $$ in domains of $\mathbb R^n$. We establish Liouville-type results in three settings. First, for globally stable solutions in $\mathbb R^n$, we prove that $u\equiv0$ if $1<p<p_...
In this paper, we focus on the existence of both global and non-global nonnegative solutions for the abstract semilinear heat equation $u_t - Lu = \sum_{i=1}^k \ell_i(t) f_i(u)$ in $L^p(X, \mu)$. Here $(X,\mu)$ represents a $\sigma$-finite measure space, $L$ is the infinitesimal generator of a sub-Markovian strongly co...
R. Castillo, Ricardo Freire, M. Loayza· 0 citations
We consider an initial value problem for a time-fractional evolution equation in a Hilbert space $X$: $$ \partial_t^{\alpha} (u(t)-a) = Au(t) \qquad \mbox{for $0<t<T$}, $$ where $\partial_t^{\alpha}$ denotes a fractional differential operator of Caputo type with order $0<\alpha<1,$ $u\colon (0,T) \to X,$ $a$ describes...
S. Chorfi, F. Golgeleyen, M. Yamamoto· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
