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

Theorem 1. If and , then any solution to problem (1)–(3) is global.
Proof. We are looking for a globally time-defined super solution in the following self-similar form

where , , ,
After some calculations we have:




Now let's show that the function is the super solution of problem (1)-(3). According to the principle of comparison, it must satisfy the following inequality:
(4)
(5)
It is easy to check that if and by the definition of B, L, D, J, inequalities (4) and (5) hold. From here and . Thus, Theorem 1 is proved by the comparison principle.
Remark 1. Theorem 1 shows that the critical exponent of the global existence of a solution to problem (1)–(3) is equal to

Theorem 2. If and , then the solution of problem (1)–(3) is unbounded in a finite time.
Proof. We will look for an unbounded sub solution in the following self-similar form:
(6)
where , and is a solution to the following problem
(7)
(8)
To be the sub solution to problem (1)-(3), the function must satisfy the following inequalities:
(9)
(10)
Let
(11)
where are constants, which are defined below. For establish the following relations


Taking the above relations to (9), (10) we obtain the inequality
in which it is easy to verify that if , then (9) and (10) hold. Thus, is the sub solution of problem (1)-(3) for any non-trivial initial data.
Theorem 3. If and , then the solution of problem (1)–(3) is unbounded in a finite time.
Proof. In this case, we will prove that, by the initial condition, the solution will be large enough, and is included in the set of initial data, for which one reaction term is enough to cause an "explosion". Consider the sub self-similar solution of problem (1)–(3) without a source:
(12)
where . Let us show that the function defined by formula (12) is the sub solution. Then, according to the comparison principle, the function must satisfy the following inequalities:
(13)
(14)
Let us set

where and are positive constants, which are defined below. After some calculations we have

Noting, that

we obtain . Calculating the inequality, we determine what makes it easy to check the validity of (13) and (14). It follows from the comparison principle that the solution to problem (1)–(3) is unbounded.