HIGHER-ORDER QUASILINEAR PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA
2006 ◽
Vol 08
(03)
◽
pp. 331-354
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Keyword(s):
As a basic model, we study the 2mth-order quasilinear parabolic equation of diffusion-absorption type [Formula: see text] where Δm,p is the 2mth-order p-Laplacian [Formula: see text]. We consider the Cauchy problem in RN × R+ with arbitrary singular initial data u0 ≠ 0 such that u0(x) = 0 for any x ≠ 0. We prove that, in the most delicate case p = q and [Formula: see text], this Cauchy problem admits the unique trivial solution u(·, t) = 0 for t > 0. For λ < λ0, such nontrivial very singular solutions are known to exist for some semilinear higher-order models. This extends the well-known result by Brezis and Friedman established in 1983 for the semilinear heat equation with p = q = m = 1.
2012 ◽
Vol 142
(2)
◽
pp. 425-448
2002 ◽
pp. 295-309
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Keyword(s):
2004 ◽
Vol 329
(1)
◽
pp. 161-196
◽
Keyword(s):
2003 ◽
Vol 182
(3)
◽
pp. 325-336
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1980 ◽
Vol 20
(5)
◽
pp. 235-241