Heat equation for an infinite bar

Let's consider an unlimited metal bar (!) assimilated to the real straight line $ℝ$ and let's call u (x,t ) the temperature of the point of abscissa x at time t, knowing that at time zero the point of abscissa x is brought to temperature h (x).

How will the bar cool down, in other words, how will u (x,t ) evolve? The mathematical modelling of this problem is the same as for the case of a finite bar [AF 141 § 5.1] , except that there are no boundary conditions. The modeling leads to the following problem: find a function...