Maximal space regularity for abstract linear non-autonomous parabolic equations

The linear non-autonomous evolution equation $\text{u′(t) ? A(t) u(t) = ?(t), t ∈ [0, T]}$, with the initial datum u(0) = x, in the space C([0, T], E), where E is a Banach space and {A(t)} is a family of infinitesimal generators of bounded analytic semigroups is considered; the domains D(A(t)) are supposed constant in t and possibly not dense in E. Maximal regularity of the strict and classical solutions, i.e., regularity of u′ and A(·)u(·) with values in the interpolation spaces D_A(0)(θ, ∞) and D_A(0)(θ) between D(A(0)) and E, is studied. A characterization of such spaces in a concrete case is also given.     

Existence and sharp regularity results for linear parabolic non-autonomous integro-differential equations

We study existence, uniqueness and maximal regularity of the strict solutionu∈C ¹([0,T],E) of the integro-differential equation \(u'(t) - A(t)u(t) - \int {_0^1 } B(t,s)u(s)ds = f(t),t \in [0,T],\) with the initial datumu(0)=x, in a Banach spaceE, {itA(itt)}_f∈|0,1| is a family of generators of analytic semigroups whose domainsD _A(t) are not constant int as well as (possibly) not dense inE, whereas {itB(itt)}_0≦11≦T is a family of closed linear operators withD _B(t,s) ?D _A(s) ∨t∈[s, T]. We prove necessary and sufficient conditions for existence of the strict solution and for Hölder continuity of its derivative; well-posedness of the problem with respect to the Hölder norms is also shown.     

Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces

In this paper we use a monotone iterative technique in the presence of the lower and upper solutions to discuss the existence of mild solutions for a class of semilinear impulsive integro-differential evolution equations of Volterra type with nonlocal conditions in a Banach space E $$\left\{ \begin{gathered} u'(t) + Au(t) = f(t,u(t),Gu(t)) t \in J,t \ne t_k , \hfill \\ \Delta _{\left. u \right|_{t = t_k } } = u\left( {t_k^ + } \right) - u\left( {t_k^ - } \right) = I_k \left( {u\left( {t_k } \right)} \right), k = 1,2, \ldots ,m, \hfill \\ u(0) = g(u) + x_0 , \hfill \\ \end{gathered} \right.$$ where A: D(A) ? E → E is a closed linear operator and ?A generates a strongly continuous semigroup T(t) (t ? 0) on E, f ∈ C(J × E × E, E), J = [0, a], 0 < t ₁ < t ₂ < ... < t _m < a, I _k ∈ C(E, E), k = 1, 2, ..., m, and g constitutes a nonlocal condition. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the extremal mild solutions between the lower and upper solutions assuming that ?A generates a compact semigroup, a strongly continuous semigroup or an equicontinuous semigroup. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Some concrete applications to partial differential equations are considered.     

Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type

In this paper, we examine, in a systematic fashion, some ill-posed problems arising in the theory of heat conduction. In abstract terms, letH be a Hilbert space andA: D (A)?H→H be an unbounded normal operator, we consider the boundary value problemü(t)=Au(t), 0<t<∞,u(0)=u ₀∈D(A), \(\mathop {\lim }\limits_{t \to 0} \left\| {u\left( t \right)} \right\| = 0\) . The problem of recoveringu ₀ whenu(T) is known for someT>0 is not well-posed. Suppose we are given approximationsx ₁,x ₂,…,x _N tou(T ₁),…,u(T _N) with 0<T, <…<T _N and positive weightsP _i,i=1,…,n, \(\sum\limits_{i = 1}^N {P_i = 1} \) such that \(Q_2 \left( {u_0 } \right) = \sum\limits_{i = 1}^N {P_i } \left\| {u\left( {T_i } \right) - x_i } \right\|^2 \leqslant \varepsilon ^2 \) . If ‖u _t(0)‖≤E for some a priori constantE, we construct a regularized solution ν(t) such that \(Q\left( {\nu \left( 0 \right)} \right) \leqslant \varepsilon ^2 \) while \(\left\| {u\left( 0 \right) - \nu \left( 0 \right)} \right\| = 0\left( {ln \left( {E/\varepsilon } \right)} \right)^{ - 1} \) and \(\left\| {u\left( t \right) - \nu \left( t \right)} \right\| = 0\left( {\varepsilon ^{\beta \left( t \right)} } \right)\) where 0<β(t)<1 and the constant in the order symbol depends uponE. The function β(t) is larger thant/m whent _k andk is the largest integer such that \((\sum\limits_{k = 1}^N {P_i (T_i )} )< (\sum\limits_{k = 1}^N {P_i (T_i )} = m\) , which β(t)=t/m on [T _k, m] and β(t)=1 on [m, ∞). Similar results are obtained if the measurement is made in the maximum norm, i.e.,Q _∞(u ₀)=max{‖u(T _i)?x _i‖, 1≤i≤N}.     

Some existence and regularity results for abstract non-autonomous parabolic equations

We study existence, uniqueness and regularity of the strict, classical and strong solutions $\text{u? C(¦0, T ¦,E)}$ of the non-autonomous evolution equation $\text{u′(t) ? A(t)u(t)=?(t)}$, with the initial datum u(0) = x, in a Banach space E, under the classical Kato-Tanabe assumptions. The domains of the operators A(t) are not needed to be dense in E. We prove necessary and sufficient conditions for existence and Hölder regularity of the solution and its derivative.     

Perturbations of Generalized Mehler Semigroups and Applications to Stochastic Heat Equations with Levy Noise and Singular Drift

In this paper we solve the Kolmogorov equation and, as a consequence, the martingale problem corresponding to a stochastic differential equation of type dX _t =AX _t dt+b(X _t )dt+dY _t , on a Hilbert space E, where (Y _t ) _t0 is a Levy process on E,A generates a C ₀-semigroup on E and b:EE. Our main point is to allow unbounded A and also singular (in particular, non-continuous) b. Our approach is based on perturbation theory of C ₀-semigroups, which we apply to generalized Mehler semigroups considered on L ²(), where is their respective invariant measure. We apply our results, in particular, to stochastic heat equations with Levy noise and singular drift.     

On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

Functional inclusion and controllability of nonlinear neutral functional differential systems

Consider the nonlinear neutral functional differential inclusion (i) $$(d/dt)D(t, x_t ) \in R(t, x_t )$$ , whereD is a continuous operator onIXC, linear inx _t , indeed of the form (4) below, with kernelD(t, ·)={0}, and atomic at 0, andR is nonempty, closed, and convex. Here,I≡[t ₀,t _I ] andC=C([-h,0],E ⁿ ). In (i), the derivative is specified in terms of the state at timet as well as the state and the derivative of the state for values oft precedingt. We use the Fan fixed-point theorem to prove the existence of a solution of (i) which satisfies two-point boundary values \(x_\omega = \phi _0 ,x_{t_1 } = \phi _1\) , where φ₀, φ₁ belong toC. We next apply this existence result to study the exact function space controllability of the neutral functional differential system (ii) $$(d/dt)D(t, x_t ) = f(t, x_t , u), u(t) \in \Omega (t, x_t )$$ . We present sufficient conditions onf and Ω which imply exact controllability between two fixed functions inC.     

Noyaux de la chaleur et estimations mixtes Lp — Lq optimales: le cas non autonome

Consider the non-autonomous initial value problem u′(t) + A(t)u(t) = f(t), u(0) = 0, where −A(t) is for each t [0,T], the generator of a bounded analytic semigroup on L²(Ω). We prove maximal L^p — L^q a priori estimates for the solution of the above equation provided the semigroups T_t are associated to kernels which satisfies an upper Gaussian bound and A(t), t [0, T] fulfills a Acquistapace-Terreni commutator condition.     

On the martingale problem for Banach space valued stochastic differential equations

LetE denote a real separable Banach space and letZ=(Z(t, f) be a family of centered, homogeneous, Gaussian independent increment processes with values inE, indexed by timet0 and the continuous functionsf:[0,t] E. If the dependence ont andf fulfills some additional properties,Z is called a gaussian random field. For continuous, adaptedE-valued processesX a stochastic integral processY = ₀ ^. Z(t, X)(dt) is defined, which is a continuous local martingale with tensor quadratic variation[Y] = ₀ ^. Q(t, X)dt, whereQ(t, f) denotes the covariance operator ofZ(t, f).Y is called a solution of the homogeneous Gaussian martingale problem, ifY = ₀ ^. Z(t, Y)(dt). Such solutions occur naturally in connection with stochastic differential equations of the type (D):dX(t)=G(t, X) dt+Z(t, X)(dt), whereG is anE-valued vector field. It is shown that a solution of (D) can be obtained by a kind of variation of parameter method, first solving a deterministic integral equation only involvingG and then solving an associated homogeneous martingale problem.     

On a smooth solution of a nonlinear periodic boundary-value problem

We establish conditions for the existence of a smooth solution of a quasilinear hyperbolic equationu _tt - u_xx = ƒ(x, t, u, u, u _x),u (0,t) = u (π,t) = 0,u (x, t+ T) = u (x, t), (x, t) ∈ [0, π] ×R, and prove a theorem on the existence and uniqueness of a solution. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1574–1576, November, 1999.     

Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions

In this paper, we are concerned with nonlocal problem for fractional evolution equations with mixed monotone nonlocal term of the form $$\left\{\begin{array}{ll}^CD^{q}_tu(t) + Au(t) = f(t, u(t), u(t)),\quad t \in J = [0, a],\\u(0) = g(u, u),\end{array}\right.$$ where E is an infinite-dimensional Banach space, \({^CD^{q}_t}\) is the Caputo fractional derivative of order \({q\in (0, 1)}\) , A : D(A) ? E → E is a closed linear operator and ?A generates a uniformly bounded C ₀-semigroup T(t) (t ≥  0) in E, \({f \in C(J\times E \times E, E)}\) , and g is appropriate continuous function so that it constitutes a nonlocal condition. Under a new concept of coupled lower and upper mild L-quasi-solutions, we construct a new monotone iterative method for nonlocal problem of fractional evolution equations with mixed monotone nonlocal term and obtain the existence of coupled extremal mild L-quasi-solutions and the mild solution between them. The results obtained generalize the recent conclusions on this topic. Finally, we present two applications to illustrate the feasibility of our abstract results.     

Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder

We study equations of the form $$\begin{gathered} u_{tt} + Lu + b(x,t)u_t = a(x,t)\left| u \right|^{\sigma - 1} u, \hfill \\ - u_t + Lu = a(x,t)\left| u \right|^{\sigma - 1} u \hfill \\ \end{gathered}$$ , whereL is a uniformly elliptic operator and 0<σ<1. In the half-cylinder II_0,∞={(x, t):x= (x ₁,...,x _n )∈ ω,t>0}, where ? ? ? _n is a bounded domain, we consider solutions satisfying the homogeneous Neumann condition forx∈?ω andt>0. We find conditions under which these solutions have compact support and prove statements of the following type: ifu(x, t)=o(t ^γ) ast→∞, then there exists aT such thatu(x, t)≡0 fort>T. In this case γ depends on the coefficients of the equation and on the exponent σ.     

Strengthening of strong and approximate convexity?

Let X be a real linear space and $D\subseteq X$ be a nonempty convex subset. Given an error function E:[0,1]×(D?D)?????{+??} and an element $t\in\left]0,1\right[$ , a function f:D??? is called (E,t)-convex if $$f(tx+(1-t)y)\le tf(x)+(1-t)f(y)+E(t,x-y)$$ for all x,y??D. The main result of this paper states that, for all a,b??(???{0})+{0,t,1?t} such that {a,b,a+b}??????, every (E,t)-convex function is also $\big(F,\frac{a}{a+b}\big)$ -convex, where $$F(s,u):=\frac{{(a+b)}^2s(1-s)}{t(1-t)}E\left(t,\frac{u}{a+b}\right),\qquad (u\in (D-D), \, s\in\left]0,1\right[).$$ As a consequence, under further assumptions on E, the strong and approximate convexity properties of (E,t)-convex functions can be strengthened.     

Brownian processes in infinite dimension

LetE be a locally convex space endowed with a centered gaussian measure . We construct a continuousE-valued brownian motionW _t with covariance . The main goal is to solve the SDE of Langevin type dX _t= dW _t–AX _t wherea andA are unbounded operators of the Cameron-Martin space of (E, ). It appears as the unique linear measurable extension of the solution of the classical Cauchy problemv(t)= u–Av(t).     

Nonlinear evolution equations in an arbitrary Banach space

This paper proves the existence of an evolution operatorU(t, s)x ₀ corresponding to a weak or generalized solution of the differential equation:du (t)/dt +A (t)u(t) ? f(t), u(s) =x ₀,t ≧ s; the operatorsA(t) are eachm-accretive in a Banach spaceX and, loosely speaking, have an “L¹ modulus of continuity” int. The continuity and differentiability properties ofU(t, s)x₀ are investigated, and some simple examples are presented.     

Existence of optimal control without convexity and a bang-bang theorem for linear volterra equations

We consider a Mayer problem of optimal control monitored by an integral equation of Volterra type: $$x(t) = x(t_1 ) + \int_{t_1 }^t { [h(t,s)x(s) + g(t, s)f(s, u(s))] ds,} $$ where the measurable control functionu satisfies a constraint of the formu(t) ∈U(t) ?E ^m,t ₁≤t≤t ₂, andg is a continuous kernel. Using the resolvent kernel associated with the kernelh, we prove the existence of an optimal usual solution for orientor fields without convexity assumptions. Further, ifU is a fixed compact set, we show the existence of an optimal bang-bang control.     

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.     

The Cauchy problem for a linear parabolic partial differential equation

Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)u_x)_x ? q(x)u = p(x)u_t, 0 < x < 1,0 < t? T; $\text{u(0, t) = ?}{}_1\text{(t), 0 < t ? T}$; $\text{u(1,t) = ?}{}_2\text{(t), 0 < t ? T}$; p(0) u_x(0, t) = g(t), 0 < t₀ ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?₁, ?₂ and g are known only approximately.     

Stability for solutions to nonlinear nonautonomous evolution equations

In this paper, we find an estimate on d(u(t), K(t)), where u is a mild solution to the nonautonomous Cauchy problem \({\dot{u}(t) + A(t)u(t) \ni 0,\, t \geq s, u(s) = u_0}\) . Here, A(t) is a family of nonlinear multivalued, ω-accretive operators in a Banach space X, with D(A(t)) possibly depending on t, and K(t) a family of closed subsets in X.