On the Cauchy problem for nonlinear parabolic equations with variable density

Abstract

We investigate the well-posedness of the Cauchy problem for a class of nonlinear parabolic equations with variable density. Sufficient conditions for uniqueness or nonuniqueness in L∞(IRN × (0, T)) (N ≥ 3) are established in dependence of the behavior of the density at infinity. We deal with conditions at infinity of Dirichlet type, and possibly inhomogeneous. © 2009 Birkhäuser Verlag Basel/Switzerland

Uniqueness and support properties of solutions to singular quasilinear parabolic equations on surfaces of revolution

“…Similar results have been proved also in [2,9,8,16,20] for the Cauchy problem in IR n , in which infinity played the role that the boundary has for problem (1.3).…”

“…Equation (1.1) can be regarded as the counterpart on a surface of revolution M of equation (1.2) which has been the object of detailed investigations (e.g., see [2,8,[15][16][17][18]20]), when is the whole of IR n or a bounded domain of IR n . It is worth to mention that it arises in situations of physical interest (see [10,22]) in a wide variety of fields involving diffusion processes.…”

Well-posedness of the Cauchy problem for nonlinear parabolic equations with variable density in the hyperbolic space

“…Moreover, for n ≥ 3, uniqueness of bounded solutions to problem (1.2), not satisfying any extra constraints at infinity, has been showed if a(x) → 0 slowly, or a(x) is bounded from below by a positive constant or it goes to infinity as |x| → ∞ (see [13,19]). …”

“…On the contrary, for n ≥ 3, existence of bounded solutions to problem (1.2), satisfying at infinity some additional conditions, has been proved, if a(x) → 0 sufficiently fast as |x| → ∞ (see [6,13,14,19,21]). Clearly, these existence results imply nonuniqueness of bounded solutions to the Cauchy problem (1.2).…”

Uniqueness and non-uniqueness of bounded solutions to singular nonlinear parabolic equations

“…Particular attention has been devoted to the Cauchy problem associated with equation (1.1) ρ ∂ t u = ∆ [G(u)] in R N × (0, T ] =: S T u = u 0 in R N × {0}; (1.2) here ρ ∈ C (R N ), u 0 ∈ L ∞ (R N ). In fact, it is well-known that problem (1.2) is well-posed in L ∞ (S T ) when N ≤ 2; moreover, it is also well-posed when N ≥ 3 and ρ(x) → 0 ''not too fast'' as |x| → ∞ (or does not vanish at all at infinity; see [2,3]). On the contrary, if N ≥ 3 and ρ(x) → 0 sufficiently fast as |x| → ∞, some constraints at infinity are needed to restore well-posedness (see [4,2,5,3,18]).…”