Assume 0 &lt; r &lt; 1 &lt; p and k , λ ≥ 0. Let the functions y , z ∈ C 1 ( 0 , T ) satisfy y ≥ 0, z &gt; 0 and the system of differential inequalities z ′ ≥ y p y ′ + λ y + k z ′ r ≥ zon ( 0 , T ). Then T &lt; ∞ ( T is the maximum lifetime of the solution).

Question

Assume   0  &amp;lt;  r  &amp;lt;  1  &amp;lt;  p and   k  ,  λ  ≥  0. Let the functions   y  ,  z  ∈      C    1    (  0  ,  T  ) satisfy   y  ≥  0,   z  &amp;gt;  0 and the system of differential inequalities      z    ′    ≥      y    p        y    ′    +  λ  y  +  k      z          ′      r        ≥  zon   (  0  ,  T  ). Then   T  &amp;lt;  ∞ (  T is the maximum lifetime of the solution).

Tucker House · Accepted Answer

By translating the origin of time, we may assume that actually   y  ,  z  ∈      C    1    (  [  0  ,  T  )  ) and   z  (  0  )  &amp;gt;  0. Fix   γ such that   max  (  r  ,  1  /  p  )  &amp;lt;  γ  &amp;lt;  1. It follows from the first inequality of system that, for all   ε  &amp;gt;  0, there exists a constant       C      ε    &amp;gt;  0 such that       C      ε                z      ′          γ    ≥      y          p          γ        +  (  3  λ  +  1  )  y  −  ε and       C      ε              z          ′          γ    ≥  3  k            z          ′        r    −  ε

Assume 0 < r < 1 < p and k , λ ≥ 0 . Let the functi

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