A group with the infinitesimal generator B
Victor A. Galaktionov, Enzo L. Mitidieri, Stanislav I. Pohozaev · 2014
Before introducing detailed spectral properties of B, we present a simple derivation of its group for proper weak solutions to be heavily used in what follows. Thus, the rescaled solution of (1), defined as w(y, τ) = t N 2mu ( yt ) , where τ = ln t ∈ IR (t > 0), (39) satisfies the necessary rescaled equation wτ = Bw (the operator B is as in (19)). (40) Then, w(y, τ) solves the CP for (40) in IRN × IR+, with data at τ = 0 (t = 1) w0(y) = u(y, 1) ≡ b(y − ·, 1) ∗ u0(·). (41) Rescaling convolution (30) yields the following explicit representation of the group with the infinitesimal generator B: w(y, τ) = eBτu(y, 1) ≡ ∫ IRN F ( y − ze− 12m τ)u0(z) dz, τ ∈ IR. (42) Performing another rescaling w(y, τ) = (1 + t) N 2mu ( y(1 + t) ) , τ = ln(1 + t) : IR+ → IR+, (43) we obtain the solution w(y, τ) of the CP for equation (40) with initial data w0(y) ≡ u0(y). Rescaling (30), we deduce a more complicated, but a standard (without the relation (41)) representation of the semigroup for τ ≥ 0, w(y, τ) = eBτu0 ≡ (1− e−τ )− N2m ∫ IRN F ( (y − ze− 12m τ )(1 − e−τ )− 12m )u0(z) dz. (44) By the Ho¨lder inequality (see e.g., (56) below), it is easy to see that w(·, τ) ∈ L2ρ(IRN ) for all τ > 0 (u0 ∈ L2ρ∗(IRN )). (45) By the descent method for constructing resolvents [105], fixing λ ∈ C, we consider an auxiliary non-homogeneous problem, wτ = Bw − eλτg for τ > 0 with w(0) = 0.