https://en.wikipedia.org/wiki/Heat_equation
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Meaning of the equation
Informally, the Laplacian operator {\displaystyle \nabla ^{2}}\nabla ^{2} gives the difference between the average value of a function in the neighborhood of a point, and its value at that point. Thus, if {\displaystyle u}u is the temperature, {\displaystyle \nabla ^{2}u}{\displaystyle \nabla ^{2}u} tells whether (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point.
By the second law of thermodynamics, heat will flow from hotter bodies to adjacent colder bodies, in proportion to the difference of temperature and of the thermal conductivity of the material between them. When heat flows into (or out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount (mass) of material, with a proportionality factor called the specific heat capacity of the material.
Therefore, the equation says that the rate {\displaystyle {\dot {u}}}{\displaystyle {\dot {u}}} at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The coefficient {\displaystyle \alpha }\alpha in the equation takes into account the thermal conductivity, the specific heat, and the density of the material.
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