Explicity resolvable equations with functional non‐linearities

Abstract

Communicated by E. de JagerWe consider the non-linear evolution equation pdu/dt + Eu+ n(u)L.(u)= h, where n(u) is a functional, and introduce assumptions which allow its explicit solution. Several concrete applications of our procedure are given.0

A nonlocal Sturm–Liouville eigenvalue problem

“…a + bi and separating the real and imaginary parts of a> gives n e) = 1 + e > -=--z -is ) -5--z. k % (y k -a)2 + b 2 k % {y k -a) 2 + b 2 Since in order for L £ to have a pair of complex conjugate eigenvalues, b must be nonzero and the imaginary part of co must vanish, one has Clearly this is impossible if c k d k does not change sign for all k. This proves the first part of the lemma. Assume now that c k d k is not of fixed sign for all k. Then there exist integers m and n such that m>n, sign (c n d n ) = -sign (c m d m ) and c k d k = 0 for any k between n and m. Then, for real X, = sign lim J ]_*•-> Vn sign lim co(X, e) = sign lim co(X, e)…”

“…Since £ x # e 2 , one has from (3.3) that y t and y 2 are two linearly independent functions, for if y 1 = ay 2 , then substituting in the equation for y 1 and comparing it with the equation for y 2 gives…”

Comparison principle for some nonlocal problems

“…f(x,y)u{y,t)dv, xedQ, t> 0; (P) Jci u(x, 0) = u0(x), x e Q, where Q is a bounded domain in R" , g(x, u) is C° in x and C1 in u with g(x, 0) = 0, and f(x , y) is a continuous function defined for x e dQ., y £ Q.. Over the last few years, many physical phenomena were formulated into nonlocal mathematical models [1,4,5,11,12,13]. However, such problems are not well studied in general.…”

The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers’ equation

“…The major difference between (P) and the model considered earlier is the use of the L" norm in (P) rather than the L2 norm. A number of authors [1,6,8,12] have investigated nonlocal problems as models for local problems. They also restricted their attention to the case q = 2.…”