Karl K. Brustad
We present an analytic approach on how to solve the problem $|\nabla u|=f(u)$, $Δu = g(u)$, in connected domains $Ω\subseteq\mathbb{R}^n$.
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In this paper we study a model for phase segregation consisting in a sistem of a partial and an ordinary differential equation. By a careful definition of maximal solution to the latter equation, this system reduces to an Allen-Cahn equation with a memory term. Global existence and uniqueness of a smooth solution are proven and a characterization of the omega-limit set is given.
On a cross-diffusion segregation problem arising from a model of interacting particles
该研究针对源自相互作用粒子模型的交叉扩散抛物型种群问题，依据扩散参数关联矩阵的正定性分情况证明解的存在性，并通过数值实验验证了强弱分离效应。
Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable coefficient are formulated. The PDE right hand sides belong to the Sobolev space $H^{-1}(Ω)$ or $\tilde H^{-1}(Ω)$, when neither classical nor canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and as well as Fredholm property and invertibility of the BDIE operators…
In this paper we prove a local-in-time existence theorem for an initial-boundary value problem related to a model of temperature-dependent phase segregation that generalizes the standard Allen-Cahn's model. The problem is ruled by a system of two differential equations, one partial the other ordinary, interpreted as balances, respectively, of microforces and of microenergy, complemented by a transcendental condition on the three unknowns, that are: the order parameter entering the standard A-C equation, the chemical potential, and the absolute temperature. The results obtained by the authors i…
Modelling pattern formation through differential repulsion
本文提出一个双物种相互作用粒子模型，旨在定量描述细胞分离现象，通过微观数学模型推导出宏观模型，并分析其参数范围，以确定分离的预期范围，同时将分析结果与数值模拟与直接粒子模拟进行比较，探讨与实验的可能联系。
We prove that for a certain class of kernels $K(y)$ that viscosity solutions of the integro-differential equation $$ \int_{\mathbb R^n} (u(x+y) - 2 u(x) + u(x-y)) K(y) dy = f(x,u(x)) $$ are locally analytic if $f$ is an analytic function. This extends the result of Albanese, Fiscella, Valdinoci that such solutions belong to certain Gevrey classes.
Hypoellipticity of analytic differential operators in general ultradifferentiable classes
研究一般超可微类中解析微分算子的亚椭圆性，通过证明解析拟微分算子等对其表现良好，研究相关正则性性质，并将Treves的刻画扩展到超可微范畴。
The mixed boundary value problem for a compressible Stokes system of partial differential equations in a bounded domain is reduced to two different systems of segregated direct Boundary Integral Equations (BDIEs) expressed in terms of surface and volume parametrix-based potential type operators. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems are proved in appropriate Sobolev spaces.
We construct of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. We show regularity properties in the frame of Schauder spaces for the corresponding single layer potentials.
We study a space-fractional diffusion problem, where the non-local diffusion flux involves the Caputo derivative of the diffusing quantity. We prove the unique existence of regular solutions to this problem by means of the semigroup theory. We show that the operator defined as divergence of product of a positive function with the Caputo derivative is a generator of analytic semigroup.
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation of differential equations of mathematical physics. Fractionality is obtained by substituting the ordinary integer-order derivative with the Caputo fractional derivative. Furthermore, operational relations between ordinary and fractional differentiation are shown and discussed in detail. Finally, a last example concerns the application of the method to the stu…
We study the real-analytic continuation of local real-analytic solutions to the Cauchy problems of quasi-linear partial differential equations of first order for a scalar function. By making use of the first integrals of the characteristic vector field and the implicit function theorem we determine the maximal domain of the analytic extension of a local solution as a single-valued function. We present some examples including the scalar conservation laws that admit global first integrals so that our method is applicable.
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