Published in
Elsevier, Journal of Mathematical Analysis and Applications, 1(308), p. 221-239, 2005
DOI: 10.1016/j.jmaa.2005.01.026
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Stability of solutions of quasilinear parabolic equations
,
Helge Holden
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Abstract
We bound the difference between solutions $u$ and $v$ of $u_t = aΔ u+\Div_x f+h$ and $v_t = bΔ v+\Div_x g+k$ with initial data $ϕ$ and $ ψ$, respectively, by $‖ u(t,⋅)-v(t,⋅)‖_{L^p(E)}\le A_E(t)‖ ϕ-ψ‖_{L^∞(\R^n)}^{2ρ_p}+ B(t)(‖ a-b‖_{∞}+ ‖ ∇_x⋅ f-∇_x⋅ g‖_{∞}+ ‖ f_u-g_u‖_{∞} + ‖ h-k‖_{∞})^{ρ_p} \abs{E}^{η_p}$. Here all functions $a$, $f$, and $h$ are smooth and bounded, and may depend on $u$, $x𝟄\R^n$, 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, $E⊂\R^n$ 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. ; Comment: 17 pages