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We study fractional parabolic equations with indefinite nonlinearities $$ frac{partial u} {partial t}(x,t) +(-Delta)^s u(x,t)= x_1 u^p(x, t),,, (x, t) in mathbb{R}^n times mathbb{R}, $$ where $0<s<1$ and $1<p<infty$. We first prove that all positive bounded solutions are monotone increasing along the $x_1$ direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce several new ideas and developed a systematic approach which may also be applied to investigate qualitative properties of solutions for many other fractional parabolic problems.
We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $Omegatimes(0,+infty)$, where $Omega$ is a bounded domain. In particular, we investigate how the behavior of the potential near
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.
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles
We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obta
We bound the difference between solutions $u$ and $v$ of $u_t = aDelta u+Div_x f+h$ and $v_t = bDelta v+Div_x g+k$ with initial data $phi$ and $ psi$, respectively, by $Vert u(t,cdot)-v(t,cdot)Vert_{L^p(E)}le A_E(t)Vert phi-psiVert_{L^infty(R^n)}^{2r