Stability of solutions of quasilinear parabolic equations

2003/06/10 by Giuseppe Maria Coclite, Helge Holden, Coclite, Giuseppe Maria +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.math/0306160

17 pages

arxiv created 2003/06/10 · arxiv updated 2009/11/30

Abstract

We bound the difference between solutions u and v of u_t = aΔu+\Div_x f+h and v_t = bΔv+\Div_x g+k with initial data ϕ and ψ, respectively, by \Vert u(t,⋅)-v(t,⋅)\Vert_L^p(E)≤ A_E(t)\Vert ϕ-ψ\Vert_L^∞(\Rⁿ)^2ρ_p+ B(t)(\Vert a-b\Vert_∞+ \Vert ∇_x⋅ f-∇_x⋅ g\Vert_∞+ \Vert f_u-g_u\Vert_∞ + \Vert h-k\Vert_∞)^ρ_p \absE^η_p. Here all functions a, f, and h are smooth and bounded, and may depend on u, x∈\Rⁿ, and t. The functions a and h may in addition depend on ∇ u. Identical assumptions hold for the functions that determine the solutions v. Furthermore, E⊂\Rⁿ is assumed to be a bounded set, and ρ_p and η_p are fractions that depend on n and p. The diffusion coefficients a and b are assumed to be strictly positive and the initial data are smooth.

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