I suddenly ran into this equation: Let $u:[a,b]\times \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying: $$\partial_t u = -u' + \frac{1}{2}u''$$ with bountary conditions $u(0,x)=g(x)$ where $g$ is a suitable function in a suitable space. Eventually, it would be really nice if $g$ can be the dirac delta.

It seems to me that one can proceed as usual using the Fourier transform and its properties as in the case of the heat equation (when the term $u'$ is gone).

Am I right? Or there is maybe a substantial difference here which I overlooked? Applying Fourier transform and assuming $u$ has compact support I got: $$\partial_t \hat{u}(t,\xi) = -\left(ik+\frac{1}{2}k^2\right)\hat{u}(t,\xi).$$

Any hints or ieas on how to get $u$ in a semiexplicit form with initial cond. $g$ or eventually $u$ being a fundamental solution?



The change $$ u(t,x)=e^{ax+bt}\,v(t,x) $$ for appropriate choice of $a,b\in\mathbb{R}$ will transform the equation into $$ v_t=v_{xx},\quad v(0,x)=e^{-ax}\,g(x). $$


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