Presentation of the Problems

Abstract

While in Parts II and III of the book we discussed the possibility to replace singular perturbation problems with the corresponding reduced models, in what follows we aim at reversing the process in the sense that we replace given problems with singularly perturbed, higher order (with respect to t) problems, admitting solutions which are more regular and approximate the solutions of the original problems. More precisely, let us consider the classical heat equation

$$ u_t- \Delta u = f\left( {x,t} \right),{\mathbf{ }}x \in \Omega ,{\mathbf{ }}0 < t < T, $$

(1)

with which we associate the Dirichlet boundary condition

$$ u\left( {x,t} \right) = 0{\mathbf{ }}for{\mathbf{ }}x{\mathbf{ }} \in \partial \Omega ,{\mathbf{ }}0 < t < T, $$

(2)

and the initial condition

$$ u\left( {x,0} \right) = u_0 \left( x \right),{\mathbf{ }}x \in \Omega , $$

(3)

where ω is a nonempty open bounded subset of ℝⁿ with smooth boundary ∂ω, and Δ is the Laplace operator with respect to x = (x₁, . . . , x_n), i.e., $\Delta u = \Sigma _{i = 1}^n u_{x_i x_i }$ . Denote by P₀ this initial-boundary value problem. If we add ±εu_tt, 0 < ε ≪ 1, to equation (E), we obtain a hyperbolic or elliptic equation, respectively,

$$ \begin{gathered} \varepsilon u_{tt}+ u_t- \Delta u = f\left( {x,t} \right),{\mathbf{ }}x \in \Omega ,{\mathbf{}}0 < t < T, \hfill \\ \varepsilon u_{tt}+ \Delta u = u_t- f\left( {x,t} \right),{\mathbf{ }}x \in \Omega ,{\mathbf{}}0 < t < T. \hfill \\ \end{gathered} $$

(4)

Now, if we associate with each of the resulting equations the original conditions (BC) and (IC), then we obtain new problems. Obviously, these problems are incomplete, since both (EE) and (HE) are of a higher order with respect to t than the original heat equation.