56,201 research outputs found
Nonuniqueness of the traveling wave speed for harmonic heat flow
No full textAbstractGiven any wave speed c∈R, we construct a traveling wave solution of ut=Δu+|∇u|2u in an infinitely long cylinder, which connects two locally stable and axially symmetric steady states at x3=±∞. Here u is a director field with values in S2⊂R3: |u|=1. The traveling wave has a singular point on the cylinder axis. In view of the bistable character of the potential, the result is surprising, and it is intimately related to the nonuniqueness of the harmonic map flow itself. We show that for only one wave speed the traveling wave behaves locally, near its singular point, as a symmetric harmonic map
A density dependent diffusion equation in population dynamics: stabilization to equilibrium
No full textWe study an evolution problem corresponding to the nonlinear diffusion equation with no flux boundary conditions. This problem has a continuum of stationary solutions. We prove the existence and uniqueness of the solution of the evolution problem and construct a Lyapunov functional in order to show that the solution stabilizes as
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
Partially overlapping travelling waves in a parabolic-hyperbolic system
No full textWe study the existence of travelling wave solutions of a one-dimensional parabolic-hyperbolic system for u (x, t) and v (x, t), which arises as a model for contact inhibition of cell growth. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and strongly parameter-dependent. In the present paper we consider a parameter regime where the minimal wave speed is positive. We show that there exists a branch of travelling wave solutions for wave speeds which are larger than the minimal one. But the main result is more surprising: for certain values of the parameters the travelling wave with minimal wave speed is not segregated (a solution is called segregated if the product uv vanishes almost everywhere) and in that case there exists a second branch of "partially overlapping" travelling wave solutions for speeds between the minimal one and that of the (unique) segregated travelling wave
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
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
Traveling wave solutions of harmonic heat flow
No full textWe prove the existence of a traveling wave solution of the equation u(t) = Delta u + vertical bar del u vertical bar(2)u in an infinitely long cylinder of radius R, which connects two locally stable and axially symmetric steady states at x(3) = +/-infinity. Here a is a director field with values in S-2 subset of R-3: vertical bar u vertical bar = 1. The traveling wave has a singular point on the cylinder axis. Letting R -> infinity we obtain a traveling wave defined in all space
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
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