Fractional boundary value problems with singularities in space variables

We are concerned with the existence of solutions for the singular fractional boundary value problem $^{c}\kern-1pt D^{\alpha}u = f(t,u)$ , u(0)+u(1)=0, u′(0)=0, where α∈(1,2), f∈C([0,1]×(??{0})) and lim _x→0 f(t,x)=∞ for all t∈[0,1]. Here, $^{c}\kern-1pt D$ is the Caputo fractional derivative. Increasing solutions of the problem vanish at points of (0,1), that is, they “pass through” the singularity of f inside of (0,1). The results are based on combining regularization and sequential techniques with a nonlinear alternative. In limit processes, the Vitali convergence theorem is used.     

Local energy decay for the wave equation on the exterior of two balls or convex bodies

An algebraic rate of decay of local energy, nonuniform with respect to the initial data, is established for solutions of the Dirichlet and Neumann problems for the scalar wave equation defined on the exterior V³ of two balls or of two convex bodies. That is, for given initial data f(x)=u(x), 0 and g(x)= u _t (x, 0), if u solves u _tt in V with either u(x, t)=0 or u _n (x,t)+(x) u(x,t,)-0 ((x)0) on V, then there exists a constant T ₀, depending upon (f, g), such that the local energy (the energy in any compact set) of u at t=T is bounded from above by QE(0)T ^–1 for TT ₀, where E(0) is the total initial energy of u and Q is a positive constant, independent of u, that depends upon V.     

The hölder continuity of generalized solutions of a class quasilinear parabolic equations

Let A cud B satisfy the Structural conditions (2), the local Hölder continuityinterior to Q=G×(0, T) is proved for the generalized solutions of quasilinearparabolic equations as follows: u₂ - divA(x, t,u,∇u) + B(x, t, u, ∇u)=0     

The approach of solutions of nonlinear diffusion equations to travelling front solutions

The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation u_t—u_xx—∞;(u)=O, x∈(—∞, ∞) , in the case ∞(0)=∞(1)=0, ∞′(0)<0, ∞′(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given.

1. Let u(x, 0)=u ₀(x) satisfy 0≦u ₀≦1. Let \(a\_ = \mathop {\lim \sup u0}\limits_{x \to - \infty } {\text{(}}x{\text{), }}\mathop {\lim \inf u0}\limits_{x \to \infty } {\text{(}}x{\text{)}}\) . Then u approaches a translate of U uniformly in x and exponentially in time, if a_? is not too far from 0, and a₊ not too far from 1.
2. Suppose \(\int\limits_{\text{0}}^{\text{1}} {f{\text{(}}u{\text{)}}du} > {\text{0}}\) . If a _? and a ₊ are not too far from 0, but u₀ exceeds a certain threshold level for a sufficiently large x-interval, then u approaches a pair of diverging travelling fronts.
3. Under certain circumstances, u approaches a “stacked” combination of wave fronts, with differing ranges.

     

Wavefronts for a cooperative tridiagonal system of differential equations

Consider the infinite system of nonlinear differential equations \(\dot u\) _n =f(_n?1, u_n, u_n+1),nε?, wherefεC ¹,D ₁ f > 0,D ₃ f>0, andf(0, 0, 0) = 0 =f(1, 1, 1). Existence of wavefronts—i.e., solutions of the formu _n (t) = U(n + ct), wherecε?,U(? ∞) = 0,U(+∞) = 1, andU is strictly increasing—is shown for functionsf which satisfy the condition: there existsa, 0<a<1, such thatf(x, x,x)<0 for 0<x<a andf(x, x, x) > 0 fora < x < 1.     

Decay estimates for some semilinear damped hyperbolic problems

Let Ω be a bounded open domain in R ⁿ , g ∶ R → R a non-decreasing continuous function such that g(0)=0 and h ε L _loc ¹ (R⁺; L ²(Ω)). Under suitable assumptions on g and h, the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u ^′′ + Lu + g(u ^′) = h(t,x) as t → + ∞. The results, obtained by use of differential inequalities, can be applied to the case of the semilinear wave equation $$u_u - \Delta u + g{\text{(}}u_t {\text{) = }}h{\text{ in }}R^ + \times \Omega ,{\text{ }}u = {\text{0 on }}R^ + \times \partial \Omega$$ in R ⁺×Ω, u=0 on R ⁺×?Ω. For instance if \(g(s) = c\left| s \right|^{p - 1} s + d\left| s \right|^{q - 1} s\) with c, d>0 and 1 < p≦q, (n?2)q≦n+2, then if \(h \in L^\infty (R + ;L^2 (\Omega ))\) , all solutions are bounded in the energy space for t≧0 and if u, v are two such solutions, the energy norm of u(t) ? v(t) decays like t ^?1/p?1 as t → + ∞.     

On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds

We consider the equation a(y)u_xx+div_y(b(y)_yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L₂() of the operatorAu= (1/a) div_y(b_yu)+(c/a) in the case ofb changing sign. This leads to the abstract problem u_xx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, u_x) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x) Ce ^-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear.     

A minimization problem related to the estimation of a lower bound for the temperature in a free boundary value problem related to the combustion of a solid

This paper investigates the least time τ_* of the first zero of the bounded solution to an initial boundary value problem for the heat equation. The heat equation is considered in the domain $$\left\{ {(x,t)| - \infty< x< s(t),0< t \leqslant T} \right\}$$ . The initial conditionu(x, 0)=φ(x) and the boundary conditionu _x (s(t),t)=?R are specified. Let τ=τ(φ,R, s) denote the first zero ofu onx=s(t), that is,u(s(τ), τ)=0. Let τ_*=min τ, where the minimum is taken over a class of functionss=s(t). The existence of τ_* is demonstrated, and a generalization of the problem is discussed.     

Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations

The aim of the paper is to give a formulation for the initial boundary value problem of parabolic-hyperbolic type in the case of nonhomogeneous boundary data a ₀. Here u=u(x,t)∈?, with (x,t)∈Q=Ω× (0,T), where Ω is a bounded domain in ? ^N with smooth boundary and T>0. The function b is assumed to be nondecreasing (allowing degeneration zones where b is constant), Φ is locally Lipschitz continuous and g∈L ^∞(Ω× (0,b)). After introducing the definition of an entropy solution to the above problem (in the spirit of Otto [14]), we prove uniqueness of the solution in the proposed setting. Moreover we prove that the entropy solution previously defined can be obtained as the limit of solutions of regularized equations of nondegenerate parabolic type (specifically the diffusion function b is approximated by functions b ^? that are strictly increasing). The approach proposed for the hyperbolic-parabolic problem can be used to prove similar results for the class of hyperbolic-elliptic boundary value problems of the form again in the case of nonconstant boundary data a ₀.     

A Locally Quadratic Glimm Functional and Sharp Convergence Rate of the Glimm Scheme for Nonlinear Hyperbolic Systems

Consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one space dimension

SBV Regularity for Hamilton–Jacobi Equations in $${{\mathbb R}^n}$$

In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations

$\partial_{t}u+H(D_{x}u)=0\quad\hbox{in }\Omega\subset{\mathbb R}\times{\mathbb R}^{n}.$

In particular, under the assumption that the Hamiltonian \({H\in C^2({\mathbb R}^n)}\) is uniformly convex, we prove that D _x u and ? _t u belong to the class SBV _loc (Ω).

     

Effects of periodic forcing on delayed bifurcations

This work is concerned with the rigorous analysis of the effects of small periodic forcing (perturbations) on the dynamical systems which present some interesting phenomena known as delayed bifurcations. We study the dynamical behavior of the system (0.1) $$\begin{gathered} \frac{{\partial u}}{{\partial t}} = f\left( {u, I_i + \varepsilon t} \right) + \varepsilon g\left( {u, I_i + \varepsilon t, \varepsilon , t} \right) \hfill \\ \left. {u\left( t \right)} \right|_{t = 0} = u_0 \left( {I_i } \right) + O\left( \varepsilon \right) \hfill \\ \end{gathered} $$ whereu ₀(I) is the solution off(u ₀(I), I)=0 andI(t)=I _i+εt is a slowly varying parameter that moves past a critical pointI_ of the system so that the linear stability aroundu ₀(I) changes from stable to unstable atI_. General results are given with respect to the effects of the perturbation εg(u,I(t),ε, t) to several important types of dynamical systems (0.2) $$\frac{{\partial u}}{{\partial t}} = f\left( {u,I_i + \varepsilon t} \right)$$ which present dynamical patterns that there exist persistent unstable solutions in the dynamical systems (delayed bifurcations) in contrast to bifurcations in the classical sense. It is shown that (1) the delayed bifurcations persist if the frequency ofg(…,…,…,t) ont is a constantΩ which is not a resonant frequency; (2) in case the frequency ofg(…,…,…,t) ont isΩ=Ω(I _i+εt) that is slowly varying, the resonance frequencies where the delayed bifurcations might be destructed are shifted downward or upward depending onΩ′(I_)>0 orΩ′(I_)<0; and (3) delayed pitchfork (simple eigenvalue) bifurcations occur in a codimension one parameter family of periodic perturbations. (1) is a rigorous analysis of the results in [3], (2) is a new and interesting phenomenon, and (3) is a generalization of the results of Diener [8] and Schecter [19].     

Phragmén-Lindelöf theorems for nonlinear elliptic equations

We obtain theorems of Phragmén-Lindelöf type for the following classes of elliptic partial differential inequalities in an arbitrary unbounded domain \(\Omega \subseteq \mathbb{R}^n ,{\text{ }}n \geqq 2\) (A.1) $$\sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial x_i }}\left( {a_{ij} 9(x)\frac{{\partial u}}{{\partial xj}}} \right)} + \sum\limits_{i = 1}^n {b_i (x,{\text{ }}u,{\text{ }}\nabla u)\frac{\partial }{{\partial x_i }}} \geqq f(x,{\text{ }}u)$$ where a _ij are elliptic in Ω and b _i ε L^∞(Ω) and where also a _ij are uniformly elliptic and Holder continuous at infinity and b _i = O(|x|⁺¹) as x → ∞; (A.2) $${\text{(A}}{\text{.2) }}\sum\limits_{i,j = 1}^n {a_{ij} (x,{\text{ }}u,{\text{ }}\nabla u)\frac{{\partial ^2 u}}{{\partial x_i \partial x_j }}} + \sum\limits_{i = 1}^n {b_i (x,{\text{ }}u,{\text{ }}\nabla u)\frac{\partial }{{\partial x_i }}} \geqq f(x,{\text{ }}u)$$ where a_ijare uniformly elliptic in Ω and b _iε L^∞(Ω); and finally (A.3) $${\text{div(}}\nabla u^p \nabla u {\text{)}} \geqq f{\text{(}}u{\text{), }}p > - 1,$$ where the operator on the left is the so-called P-Laplacian. The function f is always supposed positive and continuous. Moreover u is assumed throughout to be in the natural weak Sobolev space corresponding to the particular inequality under consideration, namely u ε. W _loc ^1,2 (Ω) ∩L _loc ^t8 (Ω) for (A.I), W _loc ^2,n(Ω) for (A.2), and W _loc ^1,p+2 (Ω) ∩ L _loc ^t8 (Ω) for (A.3). As a consequence of our results we obtain both non-existence and Liouville theorems, as well as existence theorems for (A.1).     

On the First Boundary Value Problem for Quasilinear Parabolic Equations with Two Independent Variables

This paper is concerned with the global solvability of the first initial boundary value problem for the quasilinear parabolic equations with two independent variables: a(t,x,u,u_xINF>)u_xxm u_t=f(t,x,u,u_xINF>). We investigate the case when the growth of [(|f(t,x,u,p)|)/(a(t,x,u,p))]{{|f(t,x,u,p)|}\over {a(t,x,u,p)}} with respect to p is faster than p² when |p|M X. Conditions which guarantee the global classical solvability of the problem are formulated.     

Bounded solutions of some nonlinear elliptic equations in cylindrical domains

The existence of a (unique) solution of the second-order semilinear elliptic equation $$\sum\limits_{i,j = 0}^n {a_{ij} (x)u_{x_i x_j } + f(\nabla u,u,x) = 0}$$ withx=(x ₀,x ₁,?,x _n )?(s ₀, ∞)× Ω′, for a bounded domainΩ′, together with the additional conditions $$\begin{array}{*{20}c} {u(x) = 0for(x_1 ,x_2 ,...,x_n ) \in \partial \Omega '} \\ {u(x) = \varphi (x_1 ,x_2 ,...,x_n )forx_0 = s_0 } \\ {|u(x)|globallybounded} \\ \end{array}$$ is shown to be a well-posed problem under some sign and growth restrictions off and its partial derivatives. It can be seen as an initial value problem, with initial value?, in the spaceC ₀ ⁰ $(\overline {\Omega '} )$ and satisfying the strong order-preserving property. In the case thata _ij andf do not depend onx ₀ or are periodic inx ₀, it is shown that the corresponding dynamical system has a compact global attractor. Also, conditions onf are given under which all the solutions tend to zero asx ₀ tends to infinity. Proofs are strongly based on maximum and comparison techniques.     

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x₃ and find the solution (u, p) having the following form

Contaminant Transport in Fractured Media with Sources in the Porous Domain

We study contaminant flow with sources in a fractured porous mediumconsisting of a single fracture bounded by a porous matrix. In the fracturewe assume convection, decay, surface adsorption to the interface, and lossto the porous matrix; in the porous matrix we include diffusion, decay,adsorption, and contaminant sources. The model leads to a nonhomogeneous,linear parabolic equation in a quarter-space with a differential equationfor an oblique boundary condition. Ultimately, we study the problemu _t = u _yy – u + f(x,y,t),x,y>0, t>0, u _t = –u _x + u _y – u on y = 0; u(0,0,t) =u₀(t), t>0,with zero initial data. Using Laplace transforms we obtain the Green'sfunction for the problem, and we determine how contaminant sources in theporous media are propagated in time.     

On the problem of preventing blowing-up and quenching for semilinear heat equation

In this paper,the global existence of solutions to the IVP=Δu+g(t)f(u) (t>0),u|_(t=0)=u_0(x)and the (?)PVPu_t=Δu-g(t,x)f(u)(t>0,x∈Ω),u|_(t-0)=u|_(?)(?)is investigated. As has been done in [6]the (?)duction of factor g(t) or g(t.x) innonlinear term is to prevent(?) occurrance of blowing-up or quenching of solutions.It isshown in this paper that most of the restrictions on f,g and u_0 in the theorems of[6] maybe cancelled or relaxed,that the smallness of g is required only for t large,and thatunder certain conditions controlling initial state can avoid blowing-up.     

Eventual C∞-regularity and concavity for flows in one-dimensional porous media

We study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation $$\begin{gathered} {\text{ }}u_t = \left( {u^m } \right)_{xx} {\text{ in }}Q = \mathbb{R} \times \left( {{\text{0,}}\infty } \right){\text{,}} \hfill \\ u\left( {x{\text{,0}}} \right) = u_{\text{0}} \left( x \right){\text{ for }}x \in \mathbb{R}{\text{,}} \hfill \\ \end{gathered}$$ with m > 1 and, u ₀a continuous, nonnegative function. It is well known that, across a moving interface x=ζ(t) of the solution u(x, t), the derivatives v tand v _x of the pressure v = (m/(m?1)) u ^m?1 have jump discontinuities. We prove that each moving part of the interface is a C ^∞curve and that v is C ^∞on each side of the moving interface (and up to it). We also prove that for solutions with compact support the pressure becomes a concave function of x after a finite time. This fact implies sharp convergence rates for the solution and the interfaces as t→∞.     

Dynamics as a mechanism preventing the formation of finer and finer microstructure

We study the dynamics of pattern formation in the one-dimensional partial differential equation $$u_u - (W'(u_x ))_x - u_{xxt} + u = 0{\text{ (}}u = u(x,t),{\text{ }}x \in (0,1),{\text{ }}t > 0)$$ proposed recently by Ball, Holmes, James, Pego & Swart [BHJPS] as a mathematical “cartoon” for the dynamic formation of microstructures observed in various crystalline solids. Here W is a double-well potential like 1/4((u _x )² ?1)². What makes this equation interesting and unusual is that it possesses as a Lyapunov function a free energy (consisting of kinetic energy plus a nonconvex “elastic” energy, but no interfacial energy contribution) which does not attain a minimum but favours the formation of finer and finer phase mixtures: $$E[u,u_t ] = \int\limits_0^1 {(\frac{{u_t^2 }}{2} + W(u_x ) + \frac{{u^2 }}{2})dx.}$$ Our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics, conjectured in [BHJPS] on the basis of numerical simulations of Swart & Holmes [SH]:

?While minimizing the above energy predicts infinitely fine patterns (mathematically: weak but not strong convergence of all minimizing sequences (u _nv_n) of E[u,v] in the Sobolev space W ^{1 p}(0, 1)×L²(0,1)), solutions to the evolution equation of ball et al. typically develop patterns of small but finite length scale (mathematically: strong convergence in W ₁ ^p(0,1)×L²(0,1) of all solutions (u(t),u_t(t)) with low initial energy as time t → ∞).

Moreover, in order to understand the finer details of why the dynamics fails to mimic the behaviour of minimizing sequences and how solutions select their limiting pattern, we present a detailed analysis of the evolution of a restricted class of initial data — those where the strain field u _x has a transition layer structure; our analysis includes proofs that

?at low energy, the number of phases is in fact exactly preserved, that is, there is no nucleation or coarsening

?transition layers lock in and steepen exponentially fast, converging to discontinuous stationary sharp interfaces as time t → ∞

?the limiting patterns — while not minimizing energy globally — are ‘relative minimizers’ in the weak sense of the calculus of variations, that is, minimizers among all patterns which share the same strain interface positions.