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Given the following heat equation

[tex]u_t = u_{xx} + t - x^2,[/tex]

I'd like to find all solutions [tex]u(x,t)\in C^2(\mathbb{R}^2)[/tex] such that the quotient

[tex]|u(x,t)| / (|x|^5 + |t|^5)[/tex]

goes to zero as the sum [tex]|x| + |t|[/tex] goes to infinity.

I know how to do the same problem with the usual heat equation [tex]u_t = u_{xx}[/tex], but I'm not entirely sure how to deal with this extra [tex]t - x^2[/tex] term. I suspect I can still start by taking the Fourier transform (in x) of each side to get something like:

[tex]\partial_t \hat{u}(\xi,t) = \widehat{\partial_t u}(\xi, t) = -|\xi|^2 \hat{u}(\xi,t) + \widehat{t-x^2}(\xi) = -|\xi|^2\hat{u}(\xi, t) + t\delta(\xi) - \delta^{\prime\prime}(\xi),[/tex]

which gives me an ODE in t that is easy enough to solve. The issue is that I think this method only gives me smooth (by which I mean infinitely differentiable) solutions. Are there other [tex]C^2[/tex] solutions that I am missing with this approach?

Thanks in advance for any help.

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# Modified Heat Equation

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