Heat equation with variable dissapation

I've been asked to solve the diffusion equation with variable dissipation, I have given the start of my answer but can't seem to proceed. Would appreciate a full solution as the work is due soon, thank you for any help though:

$$\frac{∂u}{∂t} - D \frac{∂^2u}{∂x^2} + e^{-pt}u = 0 , x∈(-∞,∞), t∈[0,∞]$$

subject to

$$u(x,0) = φ(x),$$

where $$D>0$$ and $$p>0$$ are given constants, and φ(x) is a given function.

I've managed to come up with this so far: Let $$u(x,t) = h(t)v(x,t)$$ then

$$h_tv + hv_t - Dhv_{xx} + e^{-pt}hv = 0$$

$$v_t - Dv_{xx} + \left(\frac{h_t}{h}+e^{-pt}\right)v = 0$$

$$v$$ will satisfy the heat equation if the third term on the LHS is zero, i.e.

$$\frac{h_t}{h} + e^{-pt} = 0$$

$$v_t - Dv_{xx} = 0$$

with initial condition $$v(x,0) = \frac{u(x,0)}{h(0)} = \frac{\varphi(x)}{h(0)}$$

• I already told you to solve for $h$ first. Have you done that? What are you still having trouble with? – Dylan Mar 4 at 4:21

@DPH24 You can just choose your $$h$$ to be a solution of the first differential equation (the one for $$h$$ above). You can choose, for example, $$h(t)=e^{e^{-pt}/p}$$ and then you solve for $$v$$ a standard IVP for the heat equation with $$v(x,0)=e^{1/p}\phi(x)$$. You get $$v(x,t)=\frac{e^{1/p}}{2\sqrt{\pi Dt}}\int\limits_{-\infty}^{+\infty}\phi(y)e^{-\frac{(x-y)^2}{4Dt}}\,dy$$ And then $$u=hv$$.