Explicit Blow-Up Time for Two Porous Medium Problems with Different Reaction Terms

Abstract

This paper deals with the blow-up phenomena of classical solutions to porous medium problems, defined in a bounded domain of $\mathbb{R}^{n}$, with n ≥ 1. We distinguish two situations: in the first case, no gradient nonlinearity is present in the reaction term contrarily to the other case. Specifically, some theoretical and general results concerning the mathematical model, existence analysis and estimates of the blow-up time t ^∗ of unbounded solutions to these problems are summarized and discussed. More exactly, for both problems, explicit lower bounds of t ^∗ if blow-up occurs are derived in the case n = 3 and in terms of an auxiliary function. On the other hand, in order to compute the real blow-up times of such blowing-up solutions and discuss their properties, a general resolution method is proposed and used in some two-dimensional examples.

Appendix

For completeness of the reader, we emphasize some details used in the proof of Theorem 3.

Proposition 1

Let the coefficients c_i (i = 1,…,6) of Theorem 3 satisfy

$$\displaystyle{ c_{3} \geq c_{5}\Big(\frac{c_{5}} {c_{6}}\Big)^{ \frac{-3\delta } {2(m+d)} }\Big( \frac{3\delta } {(2m + 2d - 3\delta )}\Big)^{- \frac{3\delta } {2(m+d)} } \frac{2(m + d)} {2(m + d) - 3\delta }. }$$

(32)

Then there exits at least a ξ ∈ (0,∞) such that

$$\displaystyle{ c_{5}\xi + c_{6}\xi ^{1-\frac{2(m+d)} {3\delta } } - c_{3} \leq 0. }$$

(33)

Proof

For any ξ ∈ (0, ∞), function $\varPhi (\xi ):= c_{5}\xi + c_{6}\xi ^{1-\frac{2(m+d)} {3\delta } }$ attains its minimum at the point

$$\displaystyle{\xi _{m} =\Big ( \frac{3\delta c_{5}} {c_{6}(2m + 2d - 3\delta )}\Big)^{ \frac{-3\delta } {2(m+d)} }.}$$

Therefore, since

$$\displaystyle{\varPhi (\xi _{m}) = c_{5}\Big(\frac{c_{5}} {c_{6}}\Big)^{ \frac{-3\delta } {2(m+d)} }\Big( \frac{3\delta } {(2m + 2d - 3\delta )}\Big)^{- \frac{3\delta } {2(m+d)} } \frac{2(m + d)} {2(m + d) - 3\delta },}$$

and (32) holds, relation (33) is proven.

In addiction, let us give this

Remark 6

Relation (32) can be explicitly written as

$$\displaystyle{ \frac{1} {k_{1}}\Big(\frac{k_{2}} {k_{1}}\Big)^{\frac{\gamma -1} {1-\mu }} \geq \frac{\varGamma ^{ \frac{3\delta } {m+d} }s^{2}(m + d)^{2}} {4m(ms - 1)} \Big[ \frac{\gamma -\mu } {\gamma -1}\Big( \frac{2\sqrt{\lambda _{1}}} {ms + q - 1}\Big)^{q}\Big]^{-\frac{\gamma -1} {1-\mu }}\Sigma, }$$

being

$$\displaystyle{ \Sigma = \frac{1-\mu } {\gamma -\mu }\Big( \frac{6(m + d)\varGamma ^{ \frac{6\alpha } {2m+3d} }d\alpha \sigma } {(2m + 3d)(2m + 2d - 3\delta )}\Big)^{1- \frac{3\delta } {2(m+d)} }. }$$

Therefore, once m, p, q and Ω are fixed in (P ₂), relation (32) is satisfied for k ₂ (respectively, k ₁) big (respectively, small) enough. Since k ₂ is the coefficient associated to − | ∇u | ^q, which contrasts the explosion, and k ₁ the one associated to u ^p, which stimulates it, this effect of coefficients k ₁ and k ₂ is coherent in terms of estimate (8). In fact, t ^∗ increases when constants c ₇ and c ₈ decreasing, which in turn decrease with k ₂ (respectively k ₁) increasing (respectively, decreasing).