By Dorina Mitrea

The idea of distributions constitutes a necessary device within the examine of partial differential equations. This textbook would provide, in a concise, mostly self-contained shape, a swift advent to the idea of distributions and its purposes to partial differential equations, together with computing basic options for the main simple differential operators: the Laplace, warmth, wave, Lame and Schrodinger operators.

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**Extra info for Distributions, Partial Differential Equations, and Harmonic Analysis (Universitext)**

**Sample text**

10) whence the former assertion of the theorem with t∗ = E 1−ν (0)/(1 − ν)C. 1] via comparison with a suitable barrier function. 1. 1) vanishes simultaneously with the source term at the same instant tf . 5. Space localization of solutions Let us denote Bρ (x0 ) = {x| |x − x0 | < ρ}, Sρ (x0 ) = ∂Bρ (x0 ), Qρ (x0 ) = Bρ (x0 ) × (0, T ). 1) in the cylinder Qρ0 (x0 ) = Bρ0 (x0 ) × (0, T ), B ρ0 ⊂ Ω, regardless of the boundary conditions on ∂Ω. 1) Bρ0 with some constant C not depending on t. By local weak solution we mean the following: 40 S.

303 (1987), 211–227. [24] N. Igbida, The mesa-limit of the porous medium equation and the Hele-Shaw problem. Diﬀ. Int. , 15 (2002), 129–146. [25] N. Igbida, The Hele Shaw Problem with Dynamical Boundary Conditions. Preprint. [26] N. Igbida, Nonlinear Heat Equation with Fast/Logarithmic Diﬀusion. Preprint. [27] N. Igbida and M. Kirane, A degenerate diﬀusion problem with dynamical boundary conditions. Math. , 323(2002), 377–396. [28] N. M. Urbano, Uniqueness for Nonlinear Degenerate Problems. NoDEA, 10 (2003), 287–307.

1) is fulﬁlled. 6) Parabolic Equations with Nonstandard Growth Conditions 39 with the exponent ν∗ (t) = np− ∗ (2 + − − − p+ ∗ σ∗ [p∗ (n + σ∗ ) − nσ∗ ] ∈ (1/2, 1). 6)) and making use of Young’s inequality, we arrive at the inequality 1 C ν∗ (t)/(2ν∗ (t)−1) E (t) + C E ν∗ (t) ≤ E ν∗ (t) + C f (·, t) 2,Ω . 3), we obtain the nonhomogeneous ordinary diﬀerential inequality 1 E (t) + C E ν∗ (t) ≤ C 2 t ε 1− tf ν 1−ν ν∗ (t) 2ν−1 2ν 2ν∗ (t)−1 . 8) can be strengthen as follows: 1 E (t) + C E ν ≤ C 2 If ε 1− t tf ν 1−ν with ν = sup ν∗ (t).