1. What is the variant to the Hardy-Littlewood-Sobolev inequality?
The variant to the Hardy-Littlewood-Sobolev inequality is a modified inequality (11) that is used to prove dynamical behaviors of solutions to equation (1). To establish this variant, several steps are required, borrowing arguments from [4]. Firstly, a variant to the HLS inequality is established using EQUATION. Then, applying the HLS inequality (34) with f = h = u and b = d - 2s, we obtain EQUATION, where Holder's inequality with 1 < 2d d+2s < m is used. This leads to the conclusion that C * <= C dsm. The existence of maximizers for the VHLS inequality can be proved by similar arguments as in [4, Lemma 3.3]. Letting 2d d+2s < m < 2 - 2s d, there exists a non-negative, radially symmetric, and non-increasing function W L 1 L m (R d) such that h(W) = C *, with normalization W L 1 (R d) = W L m (R d) = 1. Additionally, there exists R > 0 such that W solves the Euler-Lagrange equation EQUATION. The proof involves setting J(u) := h(u) u (d+2s)m-2d d(m-1) 1 u m (d-2s) d(m-1) m, and considering a maximizing sequence {u j } L 1 L m (R d) that can be assumed to be non-negative, radially symmetric, and non-increasing. The supremum can be achieved, with W satisfying the Euler-Lagrange equation, and the proof concludes with the existence of the extremal function for the modified HLS inequality.
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2. What does HLS inequality imply?
The HLS inequality implies that h(G) < . It also leads to the conclusion that J(W) = C *, resulting in W1 = Wm = 1. This inequality is derived from the given section text and plays a crucial role in the research findings. The inequality helps in establishing the relationship between various variables and their impact on the overall system. By understanding the implications of the HLS inequality, researchers can gain insights into the behavior of the system and make informed decisions based on the derived equations and properties.
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3. How can initial data bound u(x,t) L m (R d )?
Initial data can bound u(x,t) L m (R d ) by ensuring small or large values in L m norm. This is achieved through Proposition 9, which states that for any t > 0, there exists constants u1 < 1 and u2 > 1 such that the free energy solution u satisfies certain conditions. By using the variational functional and combining equations (74) and (76), it is deduced that the maximum point of g(x) is attained by u. This implies that the free energy solution u can be bounded from below or above separately, depending on the initial data. The proof also establishes the global existence and finite time blow-up of solutions to (1) in Section 4.1 and Section 4.2.
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4. Global existence proof of Theorem 3?
The global existence proof of Theorem 3 is based on the following a priori estimate for solutions to (1). After multiplying the equation by ru r-1, Young's inequality is used to arrive at EQUATION. Using condition (70), GNS inequality, and Moser iterative method, the uniformly boundedness of the weak solution is obtained. Recalling Proposition 6, it is sufficient to show that T w = by Theorem 5 to obtain a global free energy solution to (1).
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