On the properties of solutions of a nonlinear filtration problem with a source and multiple nonlinearities

Theorem 4. If and , then any solution of problem (1)-(3) is unlimited in time. Remark 2

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1. А.Алимов

Theorem 4. If and , then any solution of problem (1)-(3) is unlimited in time.
Remark 2. It is easy to see from Theorems 3 and 4 that the Fujita-type critical exponent of problem (1)-(3) is equal to
Theorem 5. If and , then the solution to problem (1)–(3) is global in time.
Theorems 4 and 5 are proved in the same way as in [13, 16].
Now let us show the asymptotics of the self-similar solution of problem (1)-(3).
Consider the following self-similar solution to problem (1)–(3):
where and , is a solution to the following problem
Consider the function
where . Let us show that function (18) is the asymptotics of solutions to problem (16), (17).
Theorem 6. Solution of problem (16), (17) with compact support has the asymptotics

when .
Proof. We will look for a solution to equation (16) in the following form
with , where for .
Substituting (19), taking into account (18), into equation (16), we obtain the form:

Note that the study of solutions of the last equation is equivalent to the study of those solutions of equation (1), each of which satisfies the inequalities on some interval :

Let us check whether the solution of Eq. (20) has a finite limit or not for . Let . Then for the derivative function we have:

We introduce an auxiliary function for analyzing solutions of the last equation:
where . From here it is easy to see that for each value of the function , the sign is preserved on a certain interval and for all one of the inequalities is fulfilled. Then, analyzing (21) using Bol's theorem [13], we conclude that the function has a limit at . It is easy to see that
at . Then, taking into account the last limit and from (20), we obtain the following algebraic equation for

From this equation we obtain that , therefore we have
in case .
Theorem 7. The solution of problem (1)-(3) has the asymptotics:

where is the above defined function.

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