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Register a new userExistence and uniqueness of solution of the differential equation describing the TASEP-LK coupled transport process
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We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, the existence of the $W^{1,2}(0,1)$ weak steady-state solution is proved by the method of upper and lower solution, and the uniqueness by a generalized maximum principle. Also, the weak steady-state solution has $C^infty$ regularity, thereby being a classical solution. We further prove the global existence and uniqueness of the time-dependent solution in $C([0,1]times [0,+infty))cap C^{2,1}([0,1]times (0,+infty))$, which, for any continuous initial value, converges to the steady-state solution uniformly in space (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species.
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This paper is devoted to the study of the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solution to the semilinear system. Next, returning to the original coordinates, we get the global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, given an admissible conservative weak solution, we find a equation to single out a unique characteristic curve through each initial point and prove the uniqueness of global admissible conservative weak solution without any additional assumptions.
For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical point for the functional on the $L^2$-normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous papers.
We provide a direct and elementary proof that the formula obtained in [MQR17] for the TASEP transition probabilities for general (one-sided) initial data solves the Kolmogorov backward equation. The same method yields the solution for the related PushASEP particle system.
In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a non-negativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many variants of Hamilton-Jacobi equations, the uniqueness for this particular type of constrained problem was not resolved, except in a few particular cases. Here, we provide a general answer to the uniqueness problem, based on three main assumptions: convexity of the Hamiltonian function H(I, x, p) with respect to p, monotonicity of H with respect to I, and BV regularity of I(t).
In this paper we prove the uniqueness of the saddle-shaped solution to the semilinear nonlocal elliptic equation $(-Delta)^gamma u = f(u)$ in $mathbb R^{2m}$, where $gamma in (0,1)$ and $f$ is of Allen-Cahn type. Moreover, we prove that this solution is stable whenever $2mgeq 14$. As a consequence of this result and the connection of the problem with nonlocal minimal surfaces, we show that the Simons cone ${(x, x) in mathbb R^{m}times mathbb R^m : |x| = |x|}$ is a stable nonlocal $(2gamma)$-minimal surface in dimensions $2mgeq 14$. Saddle-shaped solutions of the fractional Allen-Cahn equation are doubly radial, odd with respect to the Simons cone, and vanish only in this set. It was known that these solutions exist in all even dimensions and are unstable in dimensions $2$, $4$ and $6$. Thus, after our result, the stability remains an open problem only in dimensions $8$, $10$, and $12$. The importance of studying this type of solution is due to its relation with the fractional version of a conjecture by De Giorgi. Saddle-shaped solutions are the simplest non 1D candidates to be global minimizers in high dimensions, a property not yet established in any dimension.
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