Construction of the solution of an ill-posed singularly perturbed problem for the heat equation with a nonlinearityстатья
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Аннотация:From the introduction: Consider the following singularly perturbed problem
ε 2 (u t -Δu)=f(u,x,t),x∈D⊂ℝ 2 ,0<t<∞,0<ε≪1,
u| ∂D =0,u(x,0)=0,
where the nonlinearity f(u,x,t) is a continuous function and the degenerate equation f(u,x,t)=0 has roots u=ϕ i (x,t), i=1,2,⋯ We assume that the domain D has a subdomain D ' in which, as it increases, the function f(u,x,t) changes its sign from minus to plus near the even roots, i.e., f u >0 provided that the derivative f u exists. For the odd roots, the situation is opposite: the sign is changed from plus to minus, and f u <0.