Strong backward uniqueness for sublinear parabolic equations
- PDF / 404,644 Bytes
- 18 Pages / 439.37 x 666.142 pts Page_size
- 83 Downloads / 201 Views
Nonlinear Differential Equations and Applications NoDEA
Strong backward uniqueness for sublinear parabolic equations Vedansh Arya and Agnid Banerjee Abstract. In this paper, we establish strong backward uniqueness for solutions to sublinear parabolic equations of the type (1.1). The proof of our main result Theorem 1.3 is achieved by means of a new Carleman estimate and a Weiss type monotonicity formula that are tailored for such parabolic sublinear operators. Mathematics Subject Classification. 35A02, 35B60, 35K05.
Contents 1. Introduction and the statement of the main result 2. Notations and preliminaries 3. Proof of the main results Acknowledgements References
1. Introduction and the statement of the main result The purpose of this work is to establish strong backward uniqueness for parabolic sublinear equations of the type div(A(x, t)∇v) + vt + W v + h((x, t), v) = 0 in Rn × [0, T ),
(1.1)
W L∞ ≤ M,
(1.2)
where
Second author is supported in part by SERB Matrix Grant MTR/2018/000267 and by Department of Atomic Energy, Government of India, under Project No. 12-R & D-TFR5.01-0520. 0123456789().: V,-vol
50
Page 2 of 18
V. Arya and A. Banerjee
NoDEA
and the matrix A is symmetric, uniformly elliptic and satisfies λ|ξ|2 ≤ aij ξi ξj ≤ Λ|ξ|2 ∀ξ ∈ Rn .
(1.3)
Furthermore, we assume that, |∇aij (x, t)| ≤
M , |∂t aij (x, t)| ≤ M. 1 + |x|
(1.4)
On the sublinear term h, we assume the following. h ((x, t), 0) = 0, H ((x, t), s) =
(1.5) s
h(X, s)ds,
0
0 < sh ((x, t), s) ≤ qH ((x, t), s) for some q ∈ [1, 2), H((x, t), s) ≥ ε0 for all |s| > 1 and some ε0 > 0, C0 H, |∇x H| ≤ 1 + |x| |∂t H| ≤ C0 H, h ((x, t), s) ≤ C0
m
|s|pi −1 for pi s ∈ [1, 2) and some C0 > 0.
i=1
We note that from (1.5) it follows that given L > 0, there exists c0 = c0 (L) > 0, such that H((x, t), s) ≥ c0 |s|q for |s| < L.
(1.6)
A prototypical h satisfying (1.5) is given by h((x, t), v) =
l
ci (x, t)|v|pi −2 v,
i=1 C0 and |∂t ci | < C0 where for each i, pi ∈ [1, 2), 0 < k0 < ci < k1 , |∇x ci | < 1+|x| for some k0 , k1 and C0 . In this case, we can take q = max{pi }. In order to put things in the right perspective, we note that motivated by the study of nonlinear eigenvalue problems as well as the analysis of corresponding nodal domains as in [22] and also because of certain connections to porous media type equations (as in [27]), Soave and Weth in [25] established weak unique continuation for equations of the type
div(A(x)∇v) + h(x, v) + W v = 0.
(1.7)
Such equations are modeled on − Δv = |v|p−2 v.
(1.8)
Note that the study of strong unique continuation for (1.8) cannot be reduced to that for −Δ + W where the known results apply because in this case, W = |v|p−2 need not be in Lp for any p near the zero set of v as p ∈ (1, 2) and the known results
NoDEA Strong backward uniqueness for sublinear parabolic equations
Page 3 of 18
50
require W ∈ Ln/2 ( see [13]). In fact such sublinear equations have their intrinsic difficulties and this is also partly visible from the fact that the sign assumption on the sublinearity h in
Data Loading...
