The existence of strong solutions of Cauchy problem for the following evolution equation du(t)/dt + ∂1 (u (t)) - ∂2 (u(t)) ∋ f(t) is considered in a real reflexive Banach space V, where ∂1 and ∂2 are subdifferential operators from V into its dual V*. The study for this type of problems has been done by several authors in the Hilbert space setting. The scope of our study is extended to the V-V* setting. The main tool employed here is a certain approximation argument in a Hilbert space and for this purpose we need to assume that there exists a Hilbert space and for this purpose we need to assume that there exists a Hilbert space H such that V ⊂ H ≡ H* ⊂ V* with densely defined continuous injections. The applicability of our abstract framework will be exemplified in discussing the existence of solutions for the nonlinear heat equation: ut(x, t) - Δpu(x,t)- u q-2u(x,t) = f(x,t), x ∈ Ω t > 0, u/∂Ω = 0, where Ω is a bounded domain in ℝN. In particular, the existence of local (in time) weak solution is shown under the subcritical growth condition q < p* (Sobolev's critical exponent) for all data u0 ∈ W0 1,p (Ω) This fact has been conjectured but left as an open problem through many years.
- Degenerate parabolic equation
- Evolution equation
- Local existence
- Reflexive Banach space
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