Parabolic Differential Equations

For parabolic differential equations the domain G is an open set of points (t,x)=(t,x₁, ...,x _m ) with x=(x₁, ...,x _m )∈E^m,m≧1 in the m + 1dimensional space E^m+1 This notation expresses the fact that the independent variables fall into two essentially different groups. The scalar variable t is also called the “time” variable, the variable x ∈ E^m the “space” variable. We note that another n-dimensional space will occur later for parabolic systems. In order to distinguish these two spaces clearly in the notation, we shall (as in the preceding chapter) not use bold-face for the points x ∈ E^m but will use it for n-dimensional vector-functions φ=(φ₁, ...,φ _n )