86,545 research outputs found
Existence and uniqueness for a class of nonlinear elliptic equations with measure data
No full textWe study existence and uniqueness of Radon measure-valued solutions for a class of nonlinear elliptic equations in inhomogeneous media. Solutions are constructed by a regularization procedure which relies on a standard approximation of the measure data and satisfy both a persistence property and a compatibility condition prescribing the structure of their concentrated and diffuse parts with respect to a suitable capacity
Local and global solutions for a hyperbolic-elliptic model of chemotaxis on a network
No full textIn this paper, we study a hyperbolic-elliptic system on a network which arises in biological models involving chemotaxis. We also consider suitable transmission conditions at internal points of the graph which on one hand allow discontinuous density functions at nodes, and on the other guarantee the continuity of the fluxes at each node. Finally, we prove local and global existence of non-negative solutions - the latter in the case of small (in the L1-norm) initial data - as well as their uniqueness
Signed radon measure-valued solutions of flux saturated scalar conservation laws
No full textWe prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness
On a class of forward-backward parabolic equations: Properties of solutions
No full textWe study the equation ut = [φ(u)]xx + ϵ[ψ(u)]txx with suitable boundary conditions and a nonnegative Radon measure as initial datum. Here φ(0) = φ(∞) = 0, φ is increasing in (0, α) and decreasing in (α,∞), and the regularizing term ψ is increasing but bounded. It is natural to study measure-valued solutions since singularities may appear spontaneously in finite time. Positive measure-valued solutions are known to exist and to not be unique. In this paper we study qualitative properties shared by all solutions of the problem. We prove, among other things, that the singular part of a solution is nondecreasing with respect to time, so its support is nonshrinking, and, due to the possible appearance of singularities, may even expand. This phenomenon sharply distinguishes the case of bounded ψ from those of power-type ψ, where the singular part remains constant in time, and logarithmic ψ, where the singular part may grow but its support does not expand. It also distinguishes the present case from the case of φ increasing in (0, α), decreasing in (α, β), increasing in (β,∞) for some 0 < α < β < ∞, and bounded (with ψ as in this paper), where the singular part of a solution is nonincreasing in time and singularities may disappear
On a class of forward–backward parabolic equations: Existence of solutions
No full textWe study the initial-boundary value problemu(t) = [phi(u)](xx) + is an element of[psi(u)](txx) in Omega x (0, T) phi(u) + is an element of[psi(u)] t = 0 in partial derivative Omega x (0, T) u = u(0) in Omega x 0,where Omega is an interval and u0 is a nonnegative Radon measure on Omega. The map phi is increasing in (0, alpha) and decreasing in (alpha, infinity) for some alpha > 0, and satisfies phi(0) = phi(infinity) = 0. The regularizing map psi is increasing and bounded. We prove existence of suitably defined nonnegative Radon measure-valued solutions. The solution class is natural since smooth initial data may generate solutions which become measure-valued after finite time. (C) 2017 Elsevier Ltd. All rights reserved
A note on the strong maximum principle
No full textWe give a necessary and sufficient condition for the validity of the strong maximum principle in one space dimension. © 2015 Elsevier Inc
Radon measure-valued solutions of first order scalar conservation laws
No full textWe study nonnegative solutions of the Cauchy problem ∂ t u + ∂ x [ φ (u) ] = 0 in × (0, T), u = u 0 ≥ 0 in × 0 , \left\\beginaligned &\displaystyle\partial-tu+\partial-x[\varphi(u)]=0&% &\displaystyle\phantom\textin \mathbbR\times(0,T),\\ &\displaystyle u=u-0\geq 0&&\displaystyle\phantom\textin \mathbbR% \times\0\,\endaligned\right. where u 0 u-0 is a Radon measure and φ: [ 0, ∞) → \varphi\colon[0,\infty)\mapsto\mathbbR is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on φ, we prove their uniqueness if the singular part of u 0 u-0 is a finite superposition of Dirac masses. Regarding the behavior of φ at infinity, we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive waiting time (in the linear case φ (u) = u \varphi(u)=u this happens for all times). In the latter case, we describe the evolution of the singular parts
Measure-valued solutions of nonlinear parabolic equations with "logarithmic diffusion"
No full textWe prove the existence of a Radon measure-valued solution for a class of nonlinear degenerate parabolic equations with a "logarithmic diffusion" when the initial datum u (0) is a bounded Radon measure, and we study the regularity of these solutions. In particular, we prove that a regularizing effect appears if the initial datum is diffused with respect to the "C (2)-capacity" since in this case the solution becomes a summable function. Finally, we study the uniqueness of these measure-valued solutions
Pseudoparabolic regularization of forward-backward parabolic equations: a logarithmic nonlinearity
No full textWe study the initial-boundary value problem
u_t = Δφ(u) + εΔ[ψ(u)]_t in Q := Ω×(0, T],
φ(u) + ε[ψ(u)]_t = 0 in ∂Ω×(0, T],
u = u_0 ≥0 in Ω×{0},
with measure-valued initial data, assuming that the regularizing term ψ has logarithmic growth (the case of power-type ψ was dealt with in an earlier work). We prove that this case is intermediate between the case of power-type ψ and that of bounded ψ, to be addressed in a forthcoming paper. Specifically, the support of the singular part of the solution with respect to the Lebesgue measure remains constant in time (as in the case of power-type ψ), although the singular part itself need not be constant (as in the case of bounded ψ, where the support of the singular part can also increase). However, it turns out that the concentrated part of the solution with respect to the Newtonian capacity remains constant
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