Let us consider the following semilinear heat equation

together with the following conditions

and

.

This equation corresponds formally to the gradient flow associated to the energy functional

.

We prove the following

If there exists some such that then blows up in finite time.

This result is adapted from a paper by Zhong Tan published in *Commun. Partial Differential Equations* in 2001 [here].

*Proof*. Suppose and denote

.

We perform standard manipulations

and

and

.

Therefore

.

Since we have

for all . If we had , then there exists a constant such that for all there holds

,

this inequality would yield

.

On the other hand, we know that

and

and as we have for some and for all such that

.

Hence is concave on , and . This contradiction proves that .

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