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Displaying similar documents to “Determination of the unknown source term in a linear parabolic problem from the measured data at the final time”

The finite speed of propagation of solutions of the Neumann problem of a degenerate parabolic equation

Jiaqing Pan (2011)

Open Mathematics

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In this paper the finite speed of propagation of solutions and the continuous dependence on the nonlinearity of a degenerate parabolic partial differential equation are discussed. Our objective is to derive an explicit expression for the speed of propagation and the large time behavior of the solution and to show that the solution continuously depends on the nonlinearity of the equation.

Lipschitz stability in the determination of the principal part of a parabolic equation

Ganghua Yuan, Masahiro Yamamoto (2009)

ESAIM: Control, Optimisation and Calculus of Variations

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Let y ( h ) ( t , x ) be one solution to ∂ t y ( t , x ) - ∑ i , j = 1 n ∂ j ( a i j ( x ) ∂ i y ( t , x ) ) = h ( t , x ) , 0 < t < T , x ∈ Ω with a non-homogeneous term h , and y | ( 0 , T ) × ∂ Ω = 0 , where Ω ⊂ ℝ n is a bounded domain. We discuss an inverse problem of determining n ( n + 1 ) / 2 unknown functions a i j by { ∂ ν y ( h ℓ ) | ( 0 , T ) × Γ 0 , y ( h ℓ ) ( θ , · ) } 1 ≤ ℓ ≤ ℓ 0 after selecting input sources h 1 , . . . , h ℓ 0 suitably, where Γ 0 is an arbitrary subboundary, ∂ ν denotes the normal derivative, 0 < θ < T and ℓ 0 ∈ ℕ . In the case of ℓ 0 = ( n + 1 ) 2 n / 2 , we prove the Lipschitz stability in the inverse problem if we choose ( h 1 , . . . , h ℓ 0 ) from a set ℋ ⊂ { C 0 ∞ ( ( 0 , T ) × ω ) } ℓ 0 with an arbitrarily fixed subdomain ω ⊂ Ω . Moreover we can take ℓ 0 = ( n + 3 ) n / 2 by making...