The Cauchy problem for the semilinear heat equations is studied in the Orlicz space exp L 2 (R n ), where any power behavior of interaction works as a subcritical nonlinearity. We prove the existence of global solutions for the semilinear heat equations with the exponential nonlinearity under the smallness condition on the initial data in exp L 2 (R n ).
The best constant of the Sobolev inequality in the whole space is attained by the Aubin-Talenti function; however, this does not happen in bounded domains because the break in dilation invariance. In this paper, we investigate a new scale invariant form of the Sobolev inequality in a ball and show that its best constant is attained by functions of the Aubin-Talenti type. Generalization to the Caffarelli-Kohn-Nirenberg inequality in a ball is also discussed.
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