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I would want to know what is the physical interpretation of the heat equation with variable coefficients such that:

$$u_{t}-\frac{1}{1+t^2}u_{xx}=0$$

well, I think I got it, it means that the diffusivity decreases as time goes by

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I'm not quite sure, but my opinion is that it accounts for time-dependent thermal diffusivity $\alpha$ of material. In your case full heat equation would be : $$ -T^{\prime} + \alpha \nabla ^{2}T = 0$$ where $$ \alpha = \frac{1}{1+t^2} $$

Now, lets look at limit when time approaches infinity:

$$ \lim_{t \to \infty} \left(-T^{\prime} + \alpha \nabla ^{2}T\right) = -T^{\prime}$$

Thus first equation simply becomes

$$\frac {\partial T}{\partial t} = 0$$, at infinity time. This means that after long enough time there's no heat dissipation in material (no temperature change in material), because it has reached maximum heat capacity or maximum temperature in all body parts. Consider situation when you put cold metal in furnace, then heat starts to dissipate from one hot end to another cold end, until all metal rod becomes super-hot (all in red color). After this point metal rod isn't accepting anymore heat. At least there's no thermodynamic reason for a heat to travel in a rod from one end to another.

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