The Method of Introducing a Parameter in the Study of Evolutionary Equations

Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

“…Similar integral uniqueness classes for parabolic equations in unbounded domains in R d were introduced by Oleinik and Radkevich [154] and by Gushchin [91]. See also [114], [143], [101] for the results on uniqueness for positive solutions .…”

Täcklind uniqueness classes for heat equation on noncompact Riemannian manifolds

“…to the generalized solution of problem (1), (3), (4). First we observe the boundedness of the shift operators ( ) = ( + ) and Steklov's averaging operators in space˚0 ,1 ( ; Γ 1 ) under the assumptions that function ( ) is continued by zero for 0 and − ℎ :…”

“…A detailed survey of these works was provided, for instance, in [3]. It was mentioned in [4] and [5] that in the case of mixed problem, an adequate expression of a uniqueness class is that in terms of growth of the integrals…”

“…Proposition 1. Let ( ) = 0, ∈ ( ), and ( , ) be a solution to problem (1), (3), (4). Then for each > 0, ∈ (0, ], the inequality…”

The initial boundary value problem for the nonlinear parabolic equation in unbounded domain

“…In papers [8,9,17,21,24] similar results for the Cauchy problem of parabolic nonlinear equations was obtained. In [25] the authors have shown the uniqueness of a weak solution in the class of functions which do not grow faster than the function e a|x| α for |x| → ∞.…”