Admissible conditions for parabolic equations degenerating at infinity

Abstract

Abstract. Well-posedness in L ∞ (R n ) (n ≥ 3) of the Cauchy problem is studied for a class of linear parabolic equations with variable density. In view of degeneracy at infinity, some conditions at infinity are possibly needed to make the problem well-posed. Existence and uniqueness results are proved for bounded solutions that satisfy either Dirichlet or Neumann conditions at infinity.

On the Cauchy problem for nonlinear parabolic equations with variable density

“…By the same arguments used in the proof of Theorem 1.1 in [14], it is shown that This proves the result.…”

“…By uniqueness results we obtain U R = V R in Q R,T . Then, arguing as in the proof of Theorem 1.2 in [14], it is shown that…”

“…(ii) Results corresponding to Theorem 2.8, concerning nonnegative solutions of problem (1.1) with u 0 ≥ 0 were proved in [7] in the case A ≡ 0. (iii) It is known that there exists at most one solution u ∈ L ∞ (Q T ) to problem (1.1) satisfying condition (2.4) when G(u) = u (see [14]), or when u ≥ 0, u 0 ≥ 0 and A ≡ 0 (see [7]). …”

“…Observe that conditions at infinity considered in the mentioned literature are of Dirichlet type and homogeneous, for they imply that the solution goes to zero as r → ∞ in a proper sense. However, in the particular case G(u) = u, in [14] conditions at infinity of different type were considered. More precisely, existence and uniqueness of bounded classical solutions to problem (1.1) with G(u) = u, which satisfy at infinity in a suitable sense…”

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

“…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).…”