Accepted for publication in JNEEA

Existence of solutions for a class of semilinear evolution equations with impulses and delays


H. Leiva, P. Sundar

Louisiana State University, Department of Mathematics, Baton Rouge, LA -70803, USA

Received on November 12, 2016
Accepted on December 16, 2016 (with modifications)

Communicated by Toka Diagana

Abstract.  We prove the existence and uniqueness of the solutions for the followingclass of semilinear evolution equations with impulses and delays
z ́ = −Az + F(t, zt), z ∈ Z, t ∈ (0, τ ], t ≠ tk,
z(s) = φ(s), s ∈ [−r, 0],
z(tk+) = z(tk) + Jk(tk, z(tk)), k = 1, 2, 3, . . . , p.
where 0 < t1 < · · · < tp < τ , Z a Banach space,  zt defined as a function from [−r, 0] to Z by zt(s) = z(t + s), −r ≤ s ≤ 0,  and Jk : [0, τ ] × Zα → Zα, F : [0, τ ] × C(−r, 0;Zα) → Z. In the above problem,  A : D(A) ⊂ Z → Z is a sectorial operator in Z with −A being the generator of a strongly continuous compact semigroup {T(t)}t≥0, and Zα = D(Aα). The novelty of this work is that our class of evolution equations contain nonlinear terms that involve spatial derivatives. Our framework includes several important partial differential equations such as Burgers Equation with impulses and delays.
Keywords: Sectorial operator, Fractional power spaces, Semigroups, Semilinear evolution equations, Impulses, Delays, Karakostas fixed point theorem.
2010 Mathematics Subject Classification:   34K30, 34k35, 35R10; secondary: 93B05, 93C10.

This article is not yet published, it will be available for download soon.