On the existence of shock fronts for a Burgers equation with a singular source term

In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u⁰₋(x) for x < x₀, u(x, 0) = u⁰₊(x) for x > x₀, u⁰₋(x₀) > u⁰₊(x₀). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u⁰₋ and u⁰₊, a global shock front weak solution u(x, t) = u₋(x, t) for x < ϕ(t), u(x, t) = u₊(x, t) for x > ϕ(t), where u₋ and u₊ are the strong solutions corresponding (respectively) to u⁰₋ and u⁰₊ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u₋(ϕ(t), t) + u₊(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x₀. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Reaction diffusion equations with non-linear boundary conditions,blowup and steady states

In this paper we study the following problem: u_t−Δu=−f(u) in Ω×(0, T)≡Q_T, ∂u ∂n=g(u) on ∂Ω×(0, T)≡S_T, u(x, 0)=u₀(x) in Ω , where Ω⊂ℝ^N is a smooth bounded domain, f and g are smooth functions which are positive when the argument is positive, and u₀(x)>0 satisfies some smooth and compatibility conditions to guarantee the classical solution u(x, t) exists. We first obtain some existence and non-existence results for the corresponding elliptic problems. Then, we establish certain conditions for a finite time blow-up and global boundedness of the solutions of the time-dependent problem. Further, we analyse systems with same kind of boundary conditions and find some blow-up results. In the last section, we study the corresponding elliptic problems in one-dimensional domain. Our main method is the comparison principle and the construction of special forms of upper–lower solutions using related equations. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDEs

Suppose L is a second-order elliptic differential operator in ℝ^d and D is a bounded, smooth domain in ℝ^d. Let 1 < α ≤ 2 and let Γ be a closed subset of ∂D. It is known [13] that the following three properties are equivalent: (α) Γ is ∂-polar; that is, Γ is not hit by the range of the corresponding (L, α)-superdiffusion in D; (β) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where ρ(x) is the distance to the boundary and k(x, y) is the corresponding Poisson kernel; and (γ) Γ is a removable boundary singularity for the equation Lu = u^α in D; that is, if u ≥ 0 and Lu = u^α in D and if u = 0 on ∂D \ Γ, then u = 0. We investigate a similar problem for a parabolic operator in a smooth cylinder 𝒬 = ℝ₊ × D. Let Γ be a compact set on the lateral boundary of 𝒬. We show that the following three properties are equivalent: (a) Γ is 𝒢-polar; that is, Γ is not hit by the graph of the corresponding (L, α)-superdiffusion in 𝒬; (b) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where k(r, x; t, y) is the corresponding (parabolic) Poisson kernel; and (c) Γ is a removable lateral singularity for the equation u˙ + Lu = u^α in 𝒬; that is, if u ≥ 0 and u˙ + Lu = u^α in 𝒬 and if u = 0 on ∂𝒬 \ Γ and on {∞} × D, then u = 0. © 1998 John Wiley & Sons, Inc.     

Superdiffusions and removable singularities for quasilinear partial differential equations

To every second-order elliptic differential operator L and to every number α ϵ (1, 2] there is a corresponding measure-valued Markov process X called the (L, α)-superdiffusion. Suppose that Γ is a closed set in R^d. It is known that the following three statements are equivalent: (α) the range of X does not hit Γ; (β) if u ≥ 0 and Lu = u^α in R^d\Γ, then u = 0 (in other words, Γ is a removable singularity for all solutions of equation Lu = u^α); (γ) Cap_2,α′(Γ) = 0 where 1/α + 1/α′ = 1 and Cap_γ,q is the so-called Bessel capacity. The equivalence of (β) and (γ) was established by Baras and Pierre in 1984 and the equivalence of (α) and (β) was proved by Dynkin in 1991. In this paper, we consider sets Γ on the boundary ∂D of a bounded domain D and we establish (assuming that ∂D is smooth) the equivalence of the following three properties: (a) the range of X in D does not hit Γ (b) if u ≥ 0 and Lu = u^α in D, and if u → 0 as x → α ϵ ∂D\Γ, then u = 0; (c) Cap^∂_2/α,α′(Γ) = 0 where Cap^∂_γ-qis the Bessel capacity on ∂D. This implies positive answers to two conjectures posed by Dynkin a few years ago. (The conjectures have already been confirmed for α = 2 and L = Δ in a recent paper of Le Gall.) By using a combination of probabilistic and analytic arguments we not only prove the equivalence of (a)-(c) but also give a new, simplified proof of the equivalence of (α)-(γ). The paper consists of an Introduction (Section 1) and two parts, probabilistic (Sections 2 and 3) and analytic (Sections 4 and 5), that can be read independently. An important probabilistic lemma, stated in the Introduction, is proved in the Appendix. © 1996 John Wiley & Sons, Inc.     

Critical Exponents for a System of Heat Equations Coupled in a Non-linear Boundary Condition

In this paper we consider a system of heat equations u_t = Δu, v_t = Δv in an unbounded domain Ω⊂ℝ^N coupled through the Neumann boundary conditions u_v = v^p, v_v = u^q, where p>0, q>0, pq>1 and ν is the exterior unit normal on ∂Ω. It is shown that for several types of domain there exists a critical exponent such that all of positive solutions blow up in a finite time in subcritical case (including the critical case) while there exist positive global solutions in the supercritical case if initial data are small.     

An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation

We report a new unconditionally stable implicit alternating direction implicit (ADI) scheme of O(k² + h²) for the difference solution of linear hyperbolic equation u_tt + 2αu_t + β²u = u_xx + u_yy + f(x, y, t), αβ ≥ 0, 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where α > 0 and β ≥ 0 are real numbers. The resulting system of algebraic equations is solved by split method. Numerical results are provided to demonstrate the efficiency and accuracy of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 684–688, 2001     

Stability of Cahn‐Hilliard fronts

We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. © 1999 John Wiley & Sons, Inc.     

Global solutions of the Laplace equation with a nonlinear dynamical boundary condition

We study the boundedness and a priori bounds of global solutions of the problem Δu=0 in Ω×(0, T), (∂u/∂t) + (∂u/∂ν) = h(u) on ∂Ω×(0, T), where Ω is a bounded domain in ℝ^N, ν is the outer normal on ∂Ω and h is a superlinear function. As an application of our results we show the existence of sign-changing stationary solutions. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations

In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: Dαu(t)+f(t,u(t),Dμu(t))=0, u(0)=u(1)=0, where 1<α<2, 0<μ?α−1, Dα is the standard Riemann-Liouville fractional derivative, f is a positive Carathéodory function and f(t,x,y) is singular at x=0. By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques.     

Thermal avalanche for blowup solutions of semilinear heat equations

We consider the semilinear heat equation u_t = Δu + u^p both in ?^N and in a bounded domain with homogeneous Dirichlet boundary conditions, with 1 < p < p_s where p_s is the Sobolev exponent. This problem has solutions with finite‐time blowup; that is, for large enough initial data there exists T < ∞ such that u is a classical solution for 0 < t < T, while it becomes unbounded as t ↗ T. In order to understand the situation for t > T, we consider a natural approximation by reaction problems with global solution and pass to the limit. As is well‐known, the limit solution undergoes complete blowup: after it blows up at t = T, the continuation is identically infinite for all t > T. We perform here a deeper analysis of how complete blowup occurs. Actually, the singularity set of a solution that blows up as t ↗ T propagates instantaneously at time t = T to cover the whole space, producing a catastrophic discontinuity between t = T^? and t = T⁺. This is called the “avalanche.” We describe its formation as a layer appearing in the limit of the natural approximate problems. After a suitable scaling, the initial structure of the layer is given by the solution of a limit problem, described by a simple ordinary differential equation. As t proceeds past T, the solutions of the approximate problems have a traveling wave behavior with a speed that we compute. The situation in the inner region depends on the type of approximation: a conical pattern is formed with time when we approximate the power u^p by a flat truncation at level n, while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern. © 2003 Wiley Periodicals, Inc.     

Estimates on fractional power dissipative equations in function spaces

We study the fractional power dissipative equations, whose fundamental semigroup is given by e^{−t(−Δ)α} with α>0. By using an argument of duality and interpolation, we extend space-time estimates of the fractional power dissipative equations in Lebesgue spaces to the Hardy spaces and the modulation spaces. These results are substantial extensions of some known results. As applications, we study both local and global well-posedness of the Cauchy problem for the nonlinear fractional power dissipative equation u_t+(−Δ)^αu=|u|^mu for initial data in the modulation spaces.     

A characterization of periodic solutions for time‐fractional differential equations in UMD spaces and applications

We study the fractional differential equation (*) D^αu(t) + BD^βu(t) + Au(t) = f(t), 0 ? t ? 2π (0 ? β < α ? 2) in periodic Lebesgue spaces L^p(0, 2π; X) where X is a Banach space. Using functional calculus and operator valued Fourier multiplier theorems, we characterize, in UMD spaces, the well posedness of (*) in terms of R‐boundedness of the sets {(ik)^α((ik)^α + (ik)^βB + A)^?1}_{k∈ Z} and {(ik)^βB((ik)^α + (ik)^βB + A)^?1}_{k∈ Z} . Applications to the fractional problems with periodic boundary condition, which includes the time diffusion and fractional wave equations, as well as an abstract version of the Basset‐Boussinesq‐Oseen equation are treated. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ℝ⁺ (where * denotes convolution over (−∞, t)) such that u_x is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t^−α, b(t)=t^−β, and g(ξ)∼sign(ξ)∣ξ∣^γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Convexity and all‐time C∞‐regularity of the interface in flame propagation

We consider the following one‐phase free boundary problem: Find (u, Ω) such that Ω = {u > 0} and with Q_T = ?ⁿ × (0, T). Under the condition that Ω_o is convex and log u_o is concave, we show that the convexity of Ω(t) and the concavity of log u(·, t) are preserved under the flow for 0 ≤ t ≤ T as long as ?Ω(t) and u on Ω(t) are smooth. As a consequence, we show the existence of a smooth‐up‐to‐the‐interface solution, on 0 < t < T_c, with T_c denoting the extinction time of Ω(t). We also provide a new proof of a short‐time existence with C^2,α initial data on the general domain. © 2002 John Wiley & Sons, Inc.     

Convergence of Certain Functionals of Integral Fractional Processes

In this paper we consider the asymptotic behavior of functionals of processes of the form ∫ ₀ ^t u _s dB _s ^H , where B ^H is a fractional Brownian motion with Hurst parameter H, and u is a process with finite q-variation, q<1/(1−H). We establish the stable convergence of the corresponding fluctuations.     

Two‐dimensional chemotaxis models with fractional diffusion

In this paper we study the Cauchy problem for the fractional diffusion equation u_t + (?Δ)^α/2u=?·(u?(Δ^?1u)), generalizing the Keller–Segel model of chemotaxis, for the initial data u₀ in critical Besov spaces ?(?²) with r∈[1, ∞], where 1<α<2. Making use of some estimates of the linear dissipative equation in the frame of mixed time–space spaces, the Chemin ‘mono‐norm method,’ Fourier localization technique and the Littlewood–Paley theory, we obtain a local well‐posedness result. We also consider analogous ‘doubly parabolic’ models. Copyright © 2008 John Wiley & Sons, Ltd.     

Entire solutions of the KPP equation

This paper deals with the solutions defined for all time of the KPP equation u_t = u_xx + f(u),   0 < u(x,t) < 1, (x,t) ∈ ℝ², where ƒ is a KPP‐type nonlinearity defined in [0,1]: ƒ(0) = ƒ(1) = 0, ƒ′(0) > 0, ƒ′(1) < 0, ƒ > 0 in (0,1), and ƒ′(s) ≤ ƒ′(0) in [0,1]. This equation admits infinitely many traveling‐wave‐type solutions, increasing or decreasing in x. It also admits solutions that depend only on t. In this paper, we build four other manifolds of solutions: One is 5‐dimensional, one is 4‐dimensional, and two are 3‐dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling‐wave solutions are on the boundary of these four manifolds. © 1999 John Wiley & Sons, Inc.     

Certain classes of potentials for p ‐Laplacian to be non‐degenerate

Given a positive integer n and an exponent 1 ≤ α ≤ ∞. We will find explicitly the optimal bound r_n such that if the L^α norm of a potential q (t ) satisfies ‖q ‖equation/tex2gif-inf-2.gif < r_n then the n ^th Dirichlet eigenvalue of the onedimensional p ‐Laplacian with the potential q (t ): (|u ′|^{p –2} u ′)′ + (λ + q (t )) |u |^{p –2}u = 0 (1 < p < ∞) will be positive. Using these bounds, we will construct, for the Dirichlet, the Neumann, the periodic or the antiperiodic boundary conditions, certain classes of potentials q (t ) so that the p ‐Laplacian with the potential q (t ) is non‐degenerate, which means that the above equation with λ = 0 has only the trivial solution verifying the corresponding boundary condition. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Blow-up Rate for a System of Heat Equations with Non-trivial Coupling at the Boundary

We study the blow-up rate of positive radial solutions of a system of two heat equations, (u₁)_t=Δu₁(u₂)_t=Δu₂, in the ball B(0, 1), with boundary conditions Under some natural hypothesis on the matrix P=(p_ij) that guarrantee the blow-up of the solution at time T, and some assumptions of the initial data u_0i, we find that if ∥x₀∥=1 then u_i(x₀, t) goestoinfinitylike(T−t), where the α_i<0 are the solutions of (P−Id)(α₁,α₂)^t=(−1,−1)^t. As a corollary of the blow-up rate we obtain the loclaization of the blow-up set at the boundary of the domain. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Problem with periodicity conditions for a degenerating equation of mixed type

For the equation K(t)u _xx + u _tt − b ² K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| ^m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u _x (0, t) = u _x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.