Strong maximum principles for parabolic nonlinear problems with nonlocal inequalities together with integrals

Abstract

In [4] and [5], the author studied strong maximum principles for nonlinear parabolic problems with initial and nonlocal inequalities, respectively. Our purpose here is to extend results in [4] and [5] to strong maximum principles for nonlinear parabolic problems with nonlocal inequalities together with integrals. The results obtained in this paper can be applied in the theories of diffusion and heat conduction, since considered here integrals in nonlocal inequalities can be interpreted as mean amounts of the diffused substance or mean temperatures of the investigated medium.

Multiple positive solutions of parabolic systems with nonlinear, nonlocal initial conditions

“…For example, the system (1.1) with the nonlocal initial conditions (1.4) can be used to model a reaction-diffusion process of a little amount of gases in a transparent tube, in a more appropriate way than with the usual initial-value condition; for more insight on the physical interpretation of this system see the paper by Byszewski [3]. We mention that a physical motivation for the integral form of the initial condition is presented in [4,5,23]. Moreover, the system (1.1) can be used to describe the periodic solutions in the case α(u, v) = u(T ), β(u, v) = v(T ), where T > 0 and f and g are T -periodic.…”

The Non-Local Cauchy Problem for Semilinear Integrodifferential Equations With Deviating Argument

“…where −A is the infinitesimal generator of a strongly continuous semigroup T (t), t > 0, on a Banach space X, 0 t 0 < t 1 < • • • < t p t 0 +a, a > 0, u 0 ∈ X, f : [t 0 , t 0 +a]×X → X and g : [t 0 , t 0 +a] p ×X → X are given functions. Subsequently, Byszewski investigated the same type of problem for a different class of evolution equations in a Banach space [4][5][6][7][8][9][10][11]. Moreover, Corduneanu [12] and Gripenberg et al .…”

Existence of solutions of nonlinear integrodifferential equation with nonlocal condition

“…(1) u(to) + .q(tl, t2,..., tp, u( )) Uo, (2) where -A is the infinitesimal generator of a C0-semigrou p T(t), t >_ 0, on a Banach space X, 0<_t 0<tl<t2<...<tp<_t o+a,,a>O, and u 0EX and f'[t0, t 0+a]x XX, g(tl,...,tp,. )'XX are given functions.…”

“…)'XX are given functions. Subsequently, Byszewski [2][3][4][6][7][8] investigated the same type of problem stated to a different class of evolution equations in Banach space.…”