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Register a new userNonexistence of solutions to quasilinear parabolic equations with a potential in bounded domains
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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 the boundary of the domain and the power nonlinearity affect the nonexistence of solutions. Particular attention is devoted to the special case of the semilinear parabolic problem, for which we show that the critical rate of growth of the potential near the boundary ensuring nonexistence is sharp.
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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 $Vert u(t,cdot)-v(t,cdot)Vert_{L^p(E)}le A_E(t)Vert phi-psiVert_{L^infty(R^n)}^{2rho_p}+ B(t)(Vert a-bVert_{infty}+ Vert abla_xcdot f- abla_xcdot gVert_{infty}+ Vert f_u-g_uVert_{infty} + Vert h-kVert_{infty})^{rho_p} abs{E}^{eta_p}$. Here all functions $a$, $f$, and $h$ are smooth and bounded, and may depend on $u$, $xinR^n$, and $t$. The functions $a$ and $h$ may in addition depend on $ abla u$. Identical assumptions hold for the functions that determine the solutions $v$. Furthermore, $EsubsetR^n$ is assumed to be a bounded set, and $rho_p$ and $eta_p$ are fractions that depend on $n$ and $p$. The diffusion coefficients $a$ and $b$ are assumed to be strictly positive and the initial data are smooth.
We prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted volume growth conditions on geodesic balls.
In this paper we are concerned with a class of elliptic differential inequalities with a potential in bounded domains both of $mathbf{R}^m$ and of Riemannian manifolds. In particular, we investigate the effect of the behavior of the potential at the boundary of the domain on nonexistence of nonnegative solutions.
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 introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications include the parabolic maximum principle and pointwise estimates for weak solutions.
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