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Register a new userTranslating solutions for a class of quasilinear parabolic initial boundary value problems in Lorentz-Minkowski plane $mathbb{R}^{2}_{1}$
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In this paper, we investigate the evolution of spacelike curves in Lorentz-Minkowski plane $mathbb{R}^{2}_{1}$ along prescribed geometric flows (including the classical curve shortening flow or mean curvature flow as a special case), which correspond to a class of quasilinear parabolic initial boundary value problems, and can prove that this flow exists for all time. Moreover, we can also show that the evolving spacelike curves converge to a spacelike straight line or a spacelike Grim Reaper curve as time tends to infinity.
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In this paper, we consider the evolution of spacelike graphic curves defined over a piece of hyperbola $mathscr{H}^{1}(1)$, of center at origin and radius $1$, in the $2$ dimensional Lorentz-Minkowski plane $mathbb{R}^{2}_{1}$ along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic curves converge smoothly to a piece of hyperbola of center at origin and prescribed radius, which actually corresponds to a constant function defined over the piece of $mathscr{H}^{1}(1)$, as time tends to infinity.
In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $mathbb{R}^{n+1}_{1}$ along the inverse Gauss curvature flow (i.e., the evolving speed equals the $(-1/n)$-th power of the Gaussian curvature) with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of the spacelike graph of a positive constant function defined over the piece of $mathscr{H}^{n}(1)$ as time tends to infinity.
The diffusion equation is a universal and standard textbook model for partial differential equations (PDEs). In this work, we revisit its solutions, seeking, in particular, self-similar profiles. This problem connects to the classical theory of special functions and, more specifically, to the Hermite as well as the Kummer hypergeometric functions. Reconstructing the solution of the original diffusion model from novel self-similar solutions of the associated self-similar PDE, we infer that the $t^{-1/2}$ decay law of the diffusion amplitude is {it not necessary}. In particular, it is possible to engineer setups of {it both} the Cauchy problem and the initial-boundary value problem in which the solution decays at a {it different rate}. Nevertheless, we observe that the $t^{-1/2}$ rate corresponds to the dominant decay mode among integrable initial data, i.e., ones corresponding to finite mass. Hence, unless the projection to such a mode is eliminated, generically this decay will be the slowest one observed. In initial-boundary value problems, an additional issue that arises is whether the boundary data are textit{consonant} with the initial data; namely, whether the boundary data agree at all times with the solution of the Cauchy problem associated with the same initial data, when this solution is evaluated at the boundary of the domain. In that case, the power law dictated by the solution of the Cauchy problem will be selected. On the other hand, in the non-consonant cases a decomposition of the problem into a self-similar and a non-self-similar one is seen to be beneficial in obtaining a systematic understanding of the resulting solution.
It is known that for $Omega subset mathbb{R}^{2}$ an unbounded convex domain and $H>0$, there exists a graph $Gsubset mathbb{R}^{3}$ of constant mean curvature $H$ over $Omega $ with $partial G=$ $partial Omega $ if and only if $Omega $ is included in a strip of width $1/H$. In this paper we obtain results in $mathbb{H}^{2}times mathbb{R}$ in the same direction: given $Hin left( 0,1/2right) $, if $Omega $ is included in a region of $mathbb{ H}^{2}times left{ 0right} $ bounded by two equidistant hypercycles $ell(H)$ apart, we show that, if the geodesic curvature of $partial Omega $ is bounded from below by $-1,$ then there is an $H$-graph $G$ over $Omega $ with $partial G=partial Omega$. We also present more refined existence results involving the curvature of $partialOmega,$ which can also be less than $-1.$
We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is possibly time varying, non-smooth, and satisfies the exterior measure condition.
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