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We study the stability of an explicitly known, non-trivial self-similarblowup solution of the quadratic wave equation in the lowest energysupercritical dimension $d = 7$. This solution blows up at a single point andextends naturally away from the singularity. By using hyperboloidal similaritycoordinates, we prove the conditional nonlinear asymptotic stability of thissolution under small, compactly supported radial perturbations in a region ofspacetime which can be made arbitrarily close to the Cauchy horizon of thesingularity. To achieve this, we rigorously solve the underlying spectralproblem and show that the solution has exactly one genuine instability. Theunstable nature of the solution requires a careful construction of suitablyadjusted initial data at $t = 0$, which, when propagated to a family ofspacelike hypersurfaces of constant hyperboloidal time, takes the required formto guarantee convergence. By this, we introduce a new canonical method toinvestigate unstable self-similar solutions for nonlinear wave equations withinthe framework of hyperboloidal similarity coordinates.

Co-dimension one stable blowup for the quadratic wave equation beyond the light cone