In this note we establish some results of local existence and uniqueness of solutions of the equations u(x, t) = u0(x) + ∫0t u(∫0τ u(x, s)ds, τ) dτ, t ≥ 0, x ∈ ℝ u(x, t) = u0(x) + ∫0t u (1/τ ∫0τ u(x, s)ds, τ) dτ, t ≥ 0, x ∈ ℝ and u(x, t) = u0(x) + ∫0t u(∫ 0τ 1/2δ(s) ∫x-δ(s)x+δ(s) u(ε, s)dεds, τ) dτ, t ≥ 0, x ∈ ℝ, or, equivalently, for the initial value problem, respectively: ∂/∂t u(x, t) = u(∫0t u(x, s)ds, t), t ≥ 0, x ∈ ℝ u(x, 0) = u0(x), x ∈ ℝ {∂/∂t u(x, t) = u(1/t ∫0t u(x, s)ds, t), t ≥ 0, x ∈ ℝ u(x, 0) = u0(x), x ∈ ℝ and {∂/∂t u(x, t) = u(∫0t 1/2δ(s) ∫x-δ(s)x+δ(s) u(ξ, τ)dξdτ, t), t ≥ 0, x ∈ ℝ u(x, 0) = u0(x), x ∈ ℝ when u0 e δ are given function satisfying conditions. © Birkhäuser Verlag, Basel, 2006.
On a type of evolution of self-referred and hereditary phenomena
MIRANDA, Michele;
2006
Abstract
In this note we establish some results of local existence and uniqueness of solutions of the equations u(x, t) = u0(x) + ∫0t u(∫0τ u(x, s)ds, τ) dτ, t ≥ 0, x ∈ ℝ u(x, t) = u0(x) + ∫0t u (1/τ ∫0τ u(x, s)ds, τ) dτ, t ≥ 0, x ∈ ℝ and u(x, t) = u0(x) + ∫0t u(∫ 0τ 1/2δ(s) ∫x-δ(s)x+δ(s) u(ε, s)dεds, τ) dτ, t ≥ 0, x ∈ ℝ, or, equivalently, for the initial value problem, respectively: ∂/∂t u(x, t) = u(∫0t u(x, s)ds, t), t ≥ 0, x ∈ ℝ u(x, 0) = u0(x), x ∈ ℝ {∂/∂t u(x, t) = u(1/t ∫0t u(x, s)ds, t), t ≥ 0, x ∈ ℝ u(x, 0) = u0(x), x ∈ ℝ and {∂/∂t u(x, t) = u(∫0t 1/2δ(s) ∫x-δ(s)x+δ(s) u(ξ, τ)dξdτ, t), t ≥ 0, x ∈ ℝ u(x, 0) = u0(x), x ∈ ℝ when u0 e δ are given function satisfying conditions. © Birkhäuser Verlag, Basel, 2006.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
