Non-autonomous Cauchy Problems Research Articles

Well-posedness of nonautonomous abstract Cauchy problems under dissipativity conditions expressed by metric-like functionals

Under a dissipativity condition expressed by a metric-like functional, the well-posedness of a nonautonomous abstract Cauchy problem is established. The main result not only extends Martin's result on the generation of an evolution operator in a class of uniformly convex Banach spaces but also can be applied to mixed problems for nonautonomous evolution equations of Kirchhoff type with which Kato's quasilinear theory or the theory of quasi-contractive semigroups cannot directly deal. The main result also gives an affirmative answer to Kōmura's conjecture that an evolution operator of linear operators with moving domains can be generated even under a weak stability condition rather than Kato's stability condition, and the result provides an effective means for solving degenerate wave equations.

Maximal L p -regularity for evolution equations governed by non-autonomous forms in weighted spaces

We consider the maximal regularity for non-autonomous Cauchy problem u ′ ( t ) + A ( t ) u ( t ) = f ( t ) , t -a.e. , u ( 0 ) = u 0 . Each operator A ( t ) arises from a time depending on a sesquilinear form a ( t ) on a Hilbert space H with a constant domain V . We show maximal L p − regularity result in temporally weighted L p -spaces for 2 < p < ∞ and other regularity properties for the solution of the previous problem under sufficient regularity assumption on the forms and the initial value u 0 , which is significantly weaker than those from previous contributions. Our results are motivated by boundary value problems.

Perturbations of non-autonomous second-order abstract Cauchy problems

AbstractIn this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.

Well-posedness of non-autonomous transport equation on metric graphs

AbstractWe consider transport processes on metric graphs with time-dependent velocities and show that, under continuity assumption of the velocity coefficients, the corresponding non-autonomous abstract Cauchy problem is well-posed by means of evolution families and evolution semigroups.

Discrete fragmentation equations with time-dependent coefficients

We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such discrete-size fragmentation models, we allow the fragmentation coefficients to vary with time. By formulating the initial-value problem for the system as a non-autonomous abstract Cauchy problem, posed in an appropriately weighted $ \ell^1 $ space, and then applying results from the theory of evolution families, we prove the existence and uniqueness of physically relevant, classical solutions for suitably constrained coefficients.

Maximal regularity for non-autonomous Cauchy problems in weighted spaces

We consider the regularity for the non-autonomous Cauchy problem $$ u'(t) + A(t) u(t) = f(t)\quad (t \in [0, \tau]), \quad u(0) = u_0. $$ The time dependent operator A(t) is associated with (time dependent) sesquilinear forms on a Hilbert space \(\mathcal{H}\). We prove the maximal regularity result in temporally weighted L^2-spaces and other regularity properties for the solution of the problem under minimal regularity assumptions on the forms and the initial value u_0. Our results are motivated by boundary value problems.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/124/abstr.html

Maximal regularity for the damped wave equations

We consider the problem of maximal regularity for non-autonomous Cauchy problems ¨ u(t) + B(t) ˙ u(t) + A(t)u(t) = f (t) (t ∈ [0, τ ]), u(0) = u 0 , ˙ u(0) = u 1. Here, the time dependent operator A(t) is bounded from V to V ′ and B(t) is associated with a family of sesquilinear forms with domain V. We prove maximal L p-regularity results and other regularity properties for the solutions of the previous problems under minimal regularity assumptions on the operators.

Convergence Rate Estimates for Trotter Product Approximations of Solution Operators for Non-autonomous Cauchy Problems

In the present paper we advocate the Howland–Evans approach to the solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space $X$. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space $L^p(\mathcal I,X)$, $p \in \[1,\infty)$, consisting of $X$-valued functions on the time interval $\mathcal I$. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in $L^p(\mathcal I,X)$. We show that the latter also allows us to apply the full power of operator-theoretical methods to scrutinise the non-ACP, including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence.

Lions' maximal regularity problem with [formula omitted]-regularity in time

We consider the problem of maximal regularity for non-autonomous Cauchy problems{u′(t)+A(t)u(t)=f(t),t∈(0,τ]u(0)=u0. The time dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space H. We are interested in J.L. Lions's problem concerning maximal regularity of such equations. We give a positive answer to this problem under minimal regularity assumptions on the forms. Our main assumption is that the forms are piecewise H12 with respect to the variable t. This regularity assumption is optimal and our results are the most general ones on this problem.

Non-autonomous right and left multiplicative perturbations and maximal regularity

We consider the problem of maximal regularity for non-autonomous Cauchy problems u ′ (t) + B(t)A(t)u(t) + P (t)u(t) = f (t), u(0) = u 0 and u ′ (t) + A(t)B(t)u(t) + P (t)u(t) = f (t), u(0) = u 0. In both cases, the time dependent operators A(t) are associated with a family of sesquilinear forms and the multiplicative left or right perturbations B(t) as well as the additive perturbation P (t) are families of bounded operators on the considered Hilbert space. We prove maximal L p-regularity results and other regularity properties for the solutions of the previous problems under minimal regularity assumptions on the forms and perturbations.

Stability of C0-quasi semigroups in Banach spaces

We concern on the non-autonomous abstract Cauchy problems on Banach spaces X. If A(t) is the infinitesimal generator of a Co-quasi semigroup R(t, s) on X and x0 ϵ D, domain of A(t), then the solution of the equation has uniquely representation x(t) = R(0,t)x0. This representation shows that the stability of the quasi semigroup R(t, s) influences the stability of the solution. In this paper, we investigate the stabilities of C0-quasi semigroups following the existing theory of stabilities of C0-semigroups T(t) and bounded evolution operators U(t, s). We devote the uniform, exponential, and strong stability of C0-quasi semigroups in Banach spaces. The results are applicable for a large class of the time-dependent differential equations with unbounded coefficients in Banach spaces.

Non-autonomous maximal regularity for forms given by elliptic operators of bounded variation

We show maximal Lp-regularity for non-autonomous Cauchy problems provided the trace spaces are stable in some parameterized sense and the time dependence is of bounded variation. In particular on L2(Ω), for Lipschitz domains Ω and under mixed boundary conditions, we obtain maximal Lp-regularity for all p∈(1,2] for elliptic operators with coefficients aij:Ω→C satisfying aij(⋅,x)∈BV uniformly in x∈Ω.

Existence of solutions of abstract non-autonomous second order integro-differential equations

In this paper the existence of solutions of a non-autonomous abstract integro-differential equation of second order is considered. Assuming the existence of an evolution operator corresponding to the associate abstract non-autonomous Cauchy problem of second order, we establish the existence of a resolvent operator for the homogeneous integro-differential equation and the existence of mild solutions to the inhomogeneous integro-differential equation. Furthermore, we study the existence of classical solutions of the integro-differential equation. Finally, we apply our results to the study of the existence of solutions of a non-autonomous wave equation.

On evolution equations governed by non-autonomous forms

We consider a linear non-autonomous evolutionary Cauchy problem \begin{equation} \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator $A(t)$ arises from a time depending sesquilinear form $a(t,.,.)$ on a Hilbert space $H$ with constant domain $V.$ Recently a result on $L^2$-maximal regularity in $H,$ i.e., for each given $f\in L^2(0,T,H)$ and $u_0 \in V$ the problem above has a unique solution $u\in L^2(0,T,V)\cap H^1(0,T,H),$ is proved in [10] under the assumption that $a$ is symmetric and of bounded variation. The aim of this paper is to prove that the solutions of an approximate non-autonomous Cauchy problem in which $a$ is symmetric and piecewise affine are closed to the solutions of that governed by symmetric and of bounded variation form. In particular, this provide an alternative proof of the result in [10] on $L^2$-maximal regularity in $H.$

Maximal regularity for non-autonomous Robin boundary conditions

We consider a non-autonomous Cauchy problem involving linear operators associated with time-dependent forms $a(t;.,.):V\times V\to {\mathbb{C}}$ where $V$ and $H$ are Hilbert spaces such that $V$ is continuously embedded in $H$. We prove $H$-maximal regularity under a new regularity condition on the form $a$ with respect to time; namely Holder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.

Topological theory of non-autonomous parabolic evolution inclusions on a noncompact interval and applications

In this paper we consider the topological structure of the solution set of non-autonomous parabolic evolution inclusions with time delay, defined on non-compact intervals. The result restricted to compact intervals is then extended to non-autonomous parabolic control problems with time delay. Moreover, as the applications of the information about the structure, we establish the existence result of global integral solutions for non-autonomous Cauchy problems subject to nonlocal condition, and prove the invariance of a reachability set for non-autonomous control problems under single-valued nonlinear perturbations. Finally, some illustrating examples are supplied.

Singular integral operators with operator-valued kernels, and extrapolation of maximal regularity into rearrangement invariant Banach function spaces

We prove two extrapolation results for singular integral operators with operator-valued kernels, and we apply these results in order to obtain the following extrapolation of L p -maximal regularity: if an autonomous Cauchy problem on a Banach space has L p -maximal regularity for some $${p \in (1,\infty )}$$ , then it has $${\mathbb{E}_w}$$ -maximal regularity for every rearrangement invariant Banach function space $${\mathbb{E}}$$ with Boyd indices $${1 < p_\mathbb{E} \leq q_\mathbb{E} < \infty}$$ and every Muckenhoupt weight $${w \in A_{p \mathbb{E}}}$$ . We prove a similar result for nonautonomous Cauchy problems on the line.

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.

Existence of solutions of the nonautonomous abstract Cauchy problem of second order

In this paper the existence of solutions of a nonautonomous abstract Cauchy problem of second order is considered. Assuming appropriate conditions on the operator of the equation, we establish the existence of mild solutions and, in some cases, we construct an evolution operator associated to the homogeneous equation. Using this evolution operator we obtain existence of solutions for the inhomogeneous equation. Finally, we apply our results to study the existence of solutions of the nonautonomous wave equation.

Existence and flow invariance of solutions to non-autonomous Cauchy problems under separate subtangential conditions

We investigate the problem of existence and flow invariance of mild solutions to the non-autonomous Cauchy problem u̇(t)+A(t)u(t)∋0, t≥s, u(s)=u0, where A(t) is a family of nonlinear multivalued γ-accretive operators with D(A(t)) possibly depending on t.