Weak and generic bang-bang properties for continuous evolution inclusions and Baire’s method

We consider evolution inclusions, in a separable and reflexive Banach space ${\mathbb{E}}$ , of the form ${(\ast) x'(t) \in Ax(t) + F(t, x(t)), x(t_0) = c}$ and ${(**) x'(t) \in Ax(t) + {\rm ext} F(t,x(t)), x(t_0) = c}$ , where A is the infinitesimal generator of a C ₀-semigroup, F is a continuous and bounded multifunction defined on ${[t_0, t_1] \times \mathbb{E}}$ with values F(t, x) in the space of all closed convex and bounded subsets of ${\mathbb{E}}$ with nonempty interior, and ext F(t, x(t)) denotes the set of the extreme points of F(t, x(t)). For (*) and (**) we prove a weak form of the bang-bang property, namely, the closure of the set of the mild solutions of (**) contains the set of all internal solutions of (*). The proof is based on the Baire category method. This result is used to prove the following generic bang-bang property, that is, if A is the infinitesimal generator of a compact C ₀-semigroup then for most (in the sense of the Baire categories) continuous and bounded multifunctions, with closed convex and bounded values ${F(t, x) \subset \mathbb{E}}$ , the bang-bang property is actually valid, that is, the closure of the the set of the mild solutions of (**) is equal to the set of the mild solutions of (*).     

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space $\mathbb{X}$ and $f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}$ is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}_(t,s)∈Δ such that for every s∈?₊ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈?₊, and f(t,0)=0 for all t∈?₊) we can generate a (nonlinear) evolution family {X(t,s)}_(t,s)∈Δ , in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}$ is the unique solution of Eq. (4), for every $x\in\mathbb{X}$ and s∈?₊. Considering the Green’s operator $(\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds$ we prove that if the following conditions hold

• the map $\mathbb{G}{f}$ lies in $L^{q}(\mathbb{R}_{+},\mathbb{X})$ for all $f\in L^{p}(\mathbb{R}_{+},\mathbb{X})$ , and
• $\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})$ is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

Approximate Weak Invariance and Relaxation for Fully Nonlinear Differential Inclusions

In this paper we prove that a given set K is approximately weakly invariant with respect to the fully nonlinear differential inclusion $${x^\prime (t) \in Ax (t) + F (x (t))}$$ , where A is an m-dissipative operator, and F is a given multi-function in a Banach space, if and only if the set ${F(\xi)}$ is A-quasi-tangent to the set K, for every ${{\xi \in K}}$ . As an application, we establish that the approximate solutions of the given differential inclusion approximate the solutions of the relaxed (convexified) nonlinear differential inclusion $${x^\prime (t) \in Ax (t) + \overline{co}F (x (t))}$$ , with no hypotheses of Lipschitz type for multi-function F.     

Long-time asymptotics for fully nonlinear homogeneous parabolic equations

We study the long-time asymptotics of solutions of the uniformly parabolic equation $$ u_t + F(D^2u) = 0 \quad{\rm in}\, {\mathbb{R}^{n}}\times \mathbb{R}_{+},$$ for a positively homogeneous operator F, subject to the initial condition u(x, 0) =  g(x), under the assumption that g does not change sign and possesses sufficient decay at infinity. We prove the existence of a unique positive solution Φ⁺ and negative solution Φ^?, which satisfy the self-similarity relations $$\Phi^\pm (x,t) = \lambda^{\alpha^\pm}\Phi^\pm ( \lambda^{1/2} x,\lambda t ).$$ We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to ${\Phi^+}$ ( ${\Phi^-}$ ) locally uniformly in ${\mathbb{R}^{n} \times \mathbb{R}_{+}}$ . The anomalous exponents α⁺ and α^? are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in ${\mathbb{R}^{n}}$ .     

A note on the reducibility of binary affine polynomials

Stickelberger–Swan Theorem is an important tool for determining parity of the number of irreducible factors of a given polynomial. Based on this theorem, we prove in this note that every affine polynomial A(x) over ${\mathbb{F}_2}$ with degree >1, where A(x) = L(x) + 1 and ${L(x)=\sum_{i=0}^{n}{x^{2^i}}}$ is a linearized polynomial over ${\mathbb{F}_2}$ , is reducible except x ² + x + 1 and x ⁴ + x + 1. We also give some explicit factors of some special affine pentanomials over ${\mathbb{F}_2}$ .     

Fractional BVPs with strong time singularities and the limit properties of their solutions

In the first part, we investigate the singular BVP \(\tfrac{d} {{dt}}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal{H}u\) , u(0) = A, u(1) = B, ^c D ^α u(t)| _t=0 = 0, where \(\mathcal{H}\) is a continuous operator, α ∈ (0, 1) and a < 0. Here, ^c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems \(\tfrac{d} {{dt}}^c D^{\alpha _n } u + (a/t)^c D^{\alpha _n } u = f(t,u,^c D^{\beta _n } u)\) , u(0) = A, u(1) = B, \(\left. {^c D^{\alpha _n } u(t)} \right|_{t = 0} = 0\) where a < 0, 0 < β _n ≤ α _n < 1, lim _n→∞ β _n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying the boundary conditions u(0) = A, u(1) = B, u′(0) = 0.     

A generalized Cole-Hopf transformation for a two-dimensional burgers equation with a variable coefficient

The direct method is applied to the two dimensional Burgers equation with a variable coefficient (u _t + uu _x ? u _xx ) _x + s(t)u _yy = 0 is transformed into the Riccati equation $H' - \tfrac{1} {2}H^2 + \left( {\tfrac{\rho } {2} - 1} \right)H = 0$ via the ansatz $u\left( {x,y,t} \right) = \tfrac{1} {{\sqrt t }}H(\rho ) + \tfrac{y} {{2\sqrt t }}\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , provided that s(t) = t ^?3/2. Further, a generalized Cole-Hopf transformations $u\left( {x,y,t} \right) = \tfrac{y} {{2\sqrt t }} - \tfrac{2} {{\sqrt t }}\tfrac{{U_\rho (\rho ,r)}} {{U(\rho ,r)}}$ , $\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , r(t) = log t is derived to linearize (u _t + uu _x ? u _xx ) _x + t ^?3/2 u _yy to the parabolic equation $U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho$ .     

Solution dependence on initial conditions in differential variational inequalities

In the first part of this paper, we establish several sensitivity results of the solution x(t, ξ) to the ordinary differential equation (ODE) initial-value problem (IVP) dx/dt = f(x), x(0) =  ξ as a function of the initial value ξ for a nondifferentiable f(x). Specifically, we show that for $\Xi_T \equiv \{\,x(t,\xi^0): 0 \leq t \leq T\,\}$ , (a) if f is “B-differentiable” on $\Xi_T$ , then so is the solution operator x(t;·) at ξ⁰; (b) if f is “semismooth” on $\Xi_T$ , then so is x(t;·) at ξ⁰; (c) if f has a “linear Newton approximation” on $\Xi_T$ , then so does x(t;·) at ξ⁰; moreover, the linear Newton approximation of the solution operator can be obtained from the solution of a “linear” differential inclusion. In the second part of the paper, we apply these ODE sensitivity results to a differential variational inequality (DVI) and discuss (a) the existence, uniqueness, and Lipschitz dependence of solutions to subclasses of the DVI subject to boundary conditions, via an implicit function theorem for semismooth equations, and (b) the convergence of a “nonsmooth shooting method” for numerically computing such boundary-value solutions.     

Homoclinic orbits for an infinite dimensional Hamiltonian system with periodic potential

In this paper, we study the following diffusion system where $z:(u,v):\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$ , $V(x)\in C(\mathbb{R}^{N},\mathbb{R})$ is a general periodic function, g(t,x,v), f(t,x,u) are periodic in t,x and superquadratic in v,u at infinity. By using much more direct methods to prove all Cerami sequences for the energy functional are bounded and establish the existence of homoclinic orbits, which are ground state solutions for above system.     

Lower semi-continuous nonconvex perturbation of m-accrative differential inclusions in Banach spaces

In this paper we prove the existence of solutions of the differential inclusions $$\left\{ \begin{gathered} \dot X(t) \in - A_t (X(t)) + F(t,X(t)),,0 \leqslant t \leqslant T_0 \hfill \\ X(0) = x_0 \hfill \\ \end{gathered} \right.$$ whereA _t is a multivaluedm-accretive operator on a Banach spaceE andF is a measurable multifunction defined on the set \(G = \overline {\{ (t,x):A_t (x) \ne 0/\} } \) , lower semicontinuous inx and its values are not necessarily convex inE. This result generalizes some results in [1] and [9].     

Hölder estimates for trajectories of differential inclusions and HJB equations with state constraints

We consider the differential inclusion ${\dot x \in F(t, x)}$ , with F measurable in t and Lipschitz in x, together with the state constraint ${x(t) \in \Omega}$ and prove the existence of neighboring trajectories lying in the interior of Ω. Using this result, we can characterize the Value Function of the Bolza Problem as the unique lower semicontinuous solution to the corresponding Hamilton–Jacobi–Bellman equation.     

Homoclinic Solutions for a Class of Second Order Hamiltonian Systems

In this paper we consider the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system $${\ddot q}-L(t)q+\nabla W(t,q)=0, \quad\quad\quad\quad\quad\quad\quad (\rm HS)$$ where ${L\in C({\mathbb R},{\mathbb R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in {\mathbb R}}$ , W(t, q)?=?a(t)U(q) with ${a\in C({\mathbb R},{\mathbb R}^+)}$ and ${U\in C^1({\mathbb R}^n,{\mathbb R})}$ . The novelty of this paper is that, assuming L is bounded from below in the sense that there is a constant M?>?0 such that (L(t)q, q)?≥ M |q|² for all ${(t,q)\in {\mathbb R}\times {\mathbb R}^n}$ , we establish one new compact embedding theorem. Subsequently, supposing that U satisfies the global Ambrosetti–Rabinowitz condition, we obtain a new criterion to guarantee that (HS) has one nontrivial homoclinic solution using the Mountain Pass Theorem, moreover, if U is even, then (HS) has infinitely many distinct homoclinic solutions. Recent results from the literature are generalized and significantly improved.     

On large deviations for the parabolic Anderson model

The focus of this article is on the different behavior of large deviations of random functionals associated with the parabolic Anderson model above the mean versus large deviations below the mean. The functionals we treat are the solution u(x, t) to the spatially discrete parabolic Anderson model and a functional A _n which is used in analyzing the a.s. Lyapunov exponent for u(x, t). Both satisfy a “law of large numbers”, with ${\lim_{t\to \infty} \frac{1}{t} \log u(x,t)=\lambda (\kappa)}$ and ${\lim_{n\to \infty} \frac{A_n}{n}=\alpha}$ . We then think of αn and λ(κ)t as being the mean of the respective quantities A _n and log u(t, x). Typically, the large deviations for such functionals exhibits a strong asymmetry; large deviations above the mean take on a different order of magnitude from large deviations below the mean. We develop robust techniques to quantify and explain the differences.     

Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z ^d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = e^{γ t} , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = e^{γ t} , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,\eta(1))$      

Solutions of second order degenerate integro-differential equations in vector-valued function spaces

We study the well-posedness of the second order degenerate integro-differential equations(P2):(Mu)(t)+α(Mu)(t) = Au(t)+ft-∞ a(ts)Au(s)ds + f(t),0t2π,with periodic boundary conditions M u(0)=Mu(2π),(Mu)(0) =(M u)(2π),in periodic Lebesgue-Bochner spaces Lp(T,X),periodic Besov spaces B s p,q(T,X) and periodic Triebel-Lizorkin spaces F s p,q(T,X),where A and M are closed linear operators on a Banach space X satisfying D(A) D(M),a∈L1(R+) and α is a scalar number.Using known operatorvalued Fourier multiplier theorems,we completely characterize the well-posedness of(P2) in the above three function spaces.     

Spatially monotone homoclinic orbits in nonlinear parabolic equations of super-fast diffusion type

This work deals with positive classical solutions of the degenerate parabolic equation $$u_t=u^p u_{xx} \quad \quad (\star)$$ when p > 2, which via the substitution v = u ^1?p transforms into the super-fast diffusion equation ${v_t=(v^{m-1}v_x)_x}$ with ${m=-\frac{1}{p-1} \in (-1,0)}$ . It is shown that ( ${\star}$ ) possesses some entire positive classical solutions, defined for all ${t \in \mathbb {R}}$ and ${x \in \mathbb {R}}$ , which connect the trivial equilibrium to itself in the sense that u(x, t) → 0 both as t → ?∞ and as t → + ∞, locally uniformly with respect to ${x \in \mathbb {R}}$ . Moreover, these solutions have quite a simple structure in that they are monotone increasing in space. The approach is based on the construction of two types of wave-like solutions, one of them being used for ?∞ < t ≤  0 and the other one for 0 < t <  + ∞. Both types exhibit wave speeds that vary with time and tend to zero as t → ?∞ and t → + ∞, respectively. The solutions thereby obtained decay as x → ?∞, uniformly with respect to ${t \in \mathbb {R}}$ , but they are unbounded as x → + ∞. It is finally shown that within the class of functions having such a behavior as x → ?∞, there does not exist any bounded homoclinic orbit.