Initial Pointwise Bounds and Blow-up for Parabolic Choquard-Pekar Inequalities
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We study the behavior as t → 0+ of nonnegative functions (Equation presented) satisfying the parabolic Choquard-Pekar type inequalities (Equation presented) where α ∈ (0,n + 2), λ > 0, and σ ≥ 0 are constants, Φ is the heat kernel, and ∗ is the convolution operation in ℝn × (0,1). We provide optimal conditions on α, λ, and σ such that nonnegative solutions u of (0.1),(0.2) satisfy pointwise bounds in compact subsets of B1 (0) as t → 0+. We obtain similar results for nonnegative solutions of (0.1),(0.2) when Φα/n in (0.2) is replaced with the fundamental solution Φα, of the fractional heat operator (δ/δt - Δ)α/2.
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