scholarly journals Abstract Evolution Equations of Parabolic Type in Banach and Hilbert Spaces

1961 ◽  
Vol 19 ◽  
pp. 93-125 ◽  
Author(s):  
Tosio Kato
Keyword(s):  
Evolution Equation ◽  
Hilbert Spaces ◽  
Initial Value Problem ◽  
Evolution Equations ◽  
Parabolic Type ◽  
Initial Value ◽  
Abstract Evolution Equations ◽  
Uniqueness Of The Solution

The object of the present paper is to prove some theorems concerning the existence and the uniqueness of the solution of the initial value problem for the evolution equation(E).

scholarly journals On the Initial Value Problem of Stochastic Evolution Equations in Hilbert Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Xuping Zhang ◽  
Yongxiang Li ◽  
Pengyu Chen
Keyword(s):  
Mild Solution ◽  
Hilbert Spaces ◽  
Initial Value Problem ◽  
Evolution Equations ◽  
Nonlinear Term ◽  
Compact Semigroup ◽  
Growth Conditions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Initial Value

The present paper studies the initial value problem of stochastic evolution equations with compact semigroup in real separable Hilbert spaces. The existence of saturated mild solution and global mild solution is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate that our results are valuable.

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

scholarly journals On Second-Order Differential Equations with Nonsmooth Second Member

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
M. Milla Miranda ◽  
A. T. Lourêdo ◽  
L. A. Medeiros
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Hilbert Spaces ◽  
Initial Value Problem ◽  
Positive Function ◽  
Linear Operators ◽  
Initial Value ◽  
Second Order Differential Equations ◽  
Open Question ◽  
Abstract Framework

In an abstract framework, we consider the following initial value problem: u′′ + μAu + F(u)u = f  in  (0,T), u(0)=u0,u′(0)=u1, where μ is a positive function and f a nonsmooth function. Given u0, u1, and f we determine Fu in order to have a solution u of the previous equation. We analyze two cases of Fu. In our approach, we use the Theory of Linear Operators in Hilbert Spaces, the compactness Aubin-Lions Theorem, and an argument of Fixed Point. One of our two results provides an answer in a certain sense to an open question formulated by Lions in (1981, Page 284).

scholarly journals Approximate Controllability of Semilinear Impulsive Evolution Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Hugo Leiva
Keyword(s):  
Linear Operator ◽  
Evolution Equation ◽  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Smooth Functions ◽  
Bounded Linear ◽  
Unbounded Linear Operator ◽  
Approximately Controllable ◽  
Impulsive Evolution Equations

We prove the approximate controllability of the following semilinear impulsive evolution equation:z'=Az+Bu(t)+F(t,z,u), z∈Z, t∈(0,τ], z(0)=z0, z(tk+)=z(tk-)+Ik(tk,z(tk),u(tk)), k=1,2,3,…,p,where0<t1<t2<t3<⋯<tp<τ,ZandUare Hilbert spaces,u∈L2(0,τ;U),B:U→Zis a bounded linear operator,Ik,F:[0,τ]×Z×U→Zare smooth functions, andA:D(A)⊂Z→Zis an unbounded linear operator inZwhich generates a strongly continuous semigroup{T(t)}t≥0⊂Z. We suppose thatFis bounded and the linear system is approximately controllable on[0,δ]for allδ∈(0,τ). Under these conditions, we prove the following statement: the semilinear impulsive evolution equation is approximately controllable on[0,τ].

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