60 2. FOUR IMPORTANT LINEAR PDE to denote the closed circular cylinder of radius r, height r2, and top center point (x, t). Fix (x0,t0) UT and choose r 0 so small that C := C(x0,t0 r) UT . Define also the smaller cylinders C := C(x0,t0 3 4 r), C := C(x0,t0 1 2 r), which have the same top center point (x0,t0). Choose a smooth cutoff function ζ = ζ(x, t) such that 0 ζ 1, ζ 1 on C , ζ 0 near the parabolic boundary of C. Extend ζ 0 in (Rn × [0,t0]) C. 2. Assume temporarily that u C∞(UT ) and set v(x, t) := ζ(x, t)u(x, t) (x Rn, 0 t t0). Then vt = ζut + ζtu, Δv = ζΔu + 2Dζ · Du + uΔζ. Consequently (33) v = 0 on Rn × {t = 0}, and (34) vt Δv = ζtu 2Dζ · Du uΔζ =: ˜ f in Rn × (0,t0). Now set ˜(x, v t) := t 0 Rn Φ(x y, t s)˜(y, f s) dyds. According to Theorem 2 (35) ˜t v Δ˜ v = ˜ f in Rn × (0,t0) ˜ v = 0 on Rn × {t = 0}.
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