Abstract:
This paper examines nonlinear parabolic initial boundary value problems. Our focus is on generalized porous medium equations that include a variable exponent, and additional terms, which includes L1 data term. We establish the existence and uniqueness of the renormalized solution to such a class of problems under the Leray-Lions type conditions on the variable exponent elliptic operator. We assume that a nonlinear function b: ℝ → ℝ, b ∈ C1(ℝ) is a strictly increasing function such that
0 < ε ≤ b ′ ( τ ) ≤ sup b ′ ( τ ) < ∞ , τ ∈ R , b ( 0 ) = 0
The uniqueness follows from the monotony of the elliptic operator and the strict increase of the function b ∈ C1(ℝ)
Keywords:
Parabolic equation, Variable exponent Lebesgue space, Leray-Lions condition, Renormalized solution, Measure data.