Instantaneous shrinking of the support of solutions to parabolic equations with a singular absorption
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Instantaneous shrinking of the support of solutions to parabolic equations with a singular absorption Nguyen Anh Dao1 Received: 22 February 2020 / Accepted: 30 June 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract We prove a result of the instantaneous shrinking of the support of solutions to parabolic equations with a singular absorption. Our method is based on the scaling property of solutions and the strong comparison. Keywords Instantaneous shrinking of the compact support · Free boundary Mathematics Subject Classification 35K55 · 35K67 · 35K65
1 Introduction and main results In 1976, Brezis and Friedman [2] studied the obstacle problem associated to the heat equation operator: ∂t u − u = f (x, t) in R N × (0, T ), (1.1) u(x, 0) = u 0 (x) in R N . ∞ N Assume that f , f t ∈ L (R ×(0, T )), for any T > 0, and u 0 is a nonnegative measure such that R N u 0 (x)d x < ∞. Then, the authors proved an existence and uniqueness of solutions to (1.1). Moreover, if f is uniformly negative, i.e.:
f ≤ c0 < 0, in R N × (0, ∞), then the solution vanishes after a finite time; and there is a real number R > 0 such that u(x, t) = 0, ∀|x| > R, ∀t > 0, if u 0 has a compact support. In addition, if t is small enough then we have an estimate on the support as follows: S(t) ⊂ S(u 0 ) + B(0, c t| log t|),
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Nguyen Anh Dao [email protected] Institute of Applied Mathematics, University of Economics Ho Chi Minh City, Ho Chi Minh City, Vietnam 0123456789().: V,-vol
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where the constant c > 0 is independent of t, u 0 , and S(t) = x ∈ R N : u(x, t) = 0 . On the other hand, it is interesting to emphasize that u(t) is compactly supported, even u 0 need not have compact support but lim u 0 (x) = 0.
|x|→∞
(1.2)
This phenomenon is called the instantaneous shrinking of compact support (in short ISS), originally discovered in an unpublished work by L. Tartar when considering problem (1.1) with the semilinear term f (u) = −λu α , for λ > 0, and α ∈ (0, 1). After that, Evans and Knerr [12] studied (1.1) with an absorption f (u). They showed that ISS happens if (1.2) holds and f (s) : [0, ∞) → [0, ∞) is nondecreasing, such that f (0) = 0, and 1 1 ds < ∞. (1.3) √ s f (s) 0 Furthermore, there is a finite time T0 > 0 such that u(x, t) = 0, ∀(x, t) ∈ R N × (T0 , ∞). These results are still true for the degenerate equations of this type (see [4,12]) ∂t u − φ(u) + f (u) = 0 in R N × (0, T ), in R N , u(x, 0) = u 0 (x)
(1.4)
where φ is a C 1 ([0, ∞)) function, φ(0) = 0, strictly increasing, and convex such that 1 1 (1.5)
ds < ∞. 0 s f φ −1 (s) Results on ISS can be found in [15] for the porous media equation with variable coefficients. Next, we consider the parabolic p-Laplacian of this type: ∂t u − p u + f (u) = 0 in R N × (0, T ), (1.6) in R N , u(x, 0) = u 0 (x)
where p > 1, p u = div |∇u| p−2 ∇u , and f (s) : [0, ∞) → [0, ∞) is nondecreasing, such that f (0) = 0. Let u 0 satisfy (1.2). Then, Herrero [13] proved that u(t) is compactly supported for all t > 0 if either f
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