ABSTRACT
This paper deals with the propagation and vanishing properties of local weak solutions of nonlinear parabolic equations. Let Ω ⊂ R N , N = 1,2,..., be an open connected domain with the smooth boundary ∂Ω, and T > 0. We consider the problem https://www.w3.org/1998/Math/MathML"> ∂ ∂ t ( | u | α - 1 u ) = d i v ( A → ( x , t , u , ∇ u ) ) - B ( x , i , u ) + f ( x , t ) in Q = Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math1b.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> assuming that the functions https://www.w3.org/1998/Math/MathML"> A → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math55.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and B are subject to the following structural conditions: there exist constants λ > 0 and p > 1 such that https://www.w3.org/1998/Math/MathML"> ∀ ( x , t , s , ρ ) ∈ Ω × R + × R × R N M 1 | ρ | p ≤ ( A → ( x , t , s , ρ ) , ρ ) ≤ M 2 | ρ | p , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math2b.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> ∀ ( x , t , s ) ∈ Ω × R + × R s B ( x , t , s ) ≥ M 3 a ( x , t ) | s | λ + 1 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math3b.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with a(x,t) > 0 a given measurable bounded function satisfying https://www.w3.org/1998/Math/MathML"> α - 1 ∈ L ( 1 + λ ) / ( λ ˜ - λ ) ( Q ) , 0 < λ < λ ˜ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math4b.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> In (2)-(3) Mi, i = 1,2,3, are positive constants. The additional (and crucial) assumption in all further consideration is: https://www.w3.org/1998/Math/MathML"> λ < α . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math5b.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> The right-hand side f(x,t) of equation (1) and the initial data uo(x) are assumed to satisfy https://www.w3.org/1998/Math/MathML"> u 0 ∈ L α + 1 ( Ω ) , f ∈ L ( 1 + λ ) / λ ( Q ) , f a - 1 / ( 1 + λ ∼ ) ( Q ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math6b.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
